IN  MEMORIAM 
FLOR1AN  CAJORI 


SECOND  LESSONS  IN  ARITHMETIC 


AN  INTELLECTUAL  WRITTEN  ARITHMETIC 
UPON  THE  INDUCTIVE  METHOD  OF 
INSTRUCTION  AS  ILLUS- 
TRATED IN 


WARREN  COLBURN'S  FIRST  LESSONS 

BT 

H.  N.  WHEELER 


HOUGHTON,  MIFFLIN  AND  COMPANY 

Boston  :  4  Park  Street ;   New  York :  11  East  Seventeenth  Street 

>rw,  Camfcri&ge 
1892 


Copyright,  1888, 
BY  H.  N.  WHEELER. 


The  Riverside  Press,  Cambridge,  Mass.,  U.  S.  A. 
Electrotyped  and  Printed  by  H.  O.  Houghton  &  Company. 


PEEFAOE. 

SECOND  LESSONS  IN  ARITHMETIC  is  the  result 
of  an  attempt  to  prepare  a  text-book  which,  by  its 
method  of  developing  the  mind  of  the  learner,  by 
the  emphasis  that  it  places  on  fundamental  princi- 
ples, and  by  the  omission  of  useless  subjects  and 
arithmetical  terms  known  only  in  the  school-room, 
will  meet  the  wants  of  those  teachers  and  busi- 
ness men  throughout  the  United  States  who  de- 
mand that  the  essentials  of  Arithmetic  shall  be 
better  taught  than  heretofore,  and  that  the  non- 
essentials  shall  be  omitted. 

I  believe  that  there  can  be  no  better  prepara- 
tion either  for  the  business  of  life  or  for  advanced 
intellectual  work  than  a  thorough  understanding 
of  the  fundamental  principles  of  Addition,  Sub- 
traction, Multiplication,  Division,  Fractions,  Deci- 
mals, and  Percentage  (including  Interest),  coupled 
with  a  self-reliant  power  of  analysis  sufficient  to 
enable  one  to  deduce  what  is  required  from  what 
is  given  by  the  aid  of  his  own  understanding 
rather  than  by  the  aid  of  his  memory  of  rules  or 


iv  Preface. 

methods.  The  subjects  just  mentioned  I  consider 
to  be  the  essentials  of  Arithmetic ;  everything  else 
in  this  book  has  been  brought  in  for  the  sake  of 
illustration.  Such  illustrations l  have  been  chosen, 
however,  as  will  appeal  to  the  intelligence  of  the 
pupil  and  give  him  a  broader  experience  in  those 
subjects  to  which  in  after  life  he  will  be  called 
upon  to  apply  his  arithmetical  powers. 

I  have  tried  in  all  cases  to  follow  the  Inductive 
Method  of  Instruction  as  illustrated  in  Warren 
Colburn's  First  Lessons,  a  book  which  has  con- 
fessedly done  more  for  the  cause  of  education 
than  any  other  text-book  that  has  ever  been  pub- 
lished. The  Inductive-Colburn  method,  as  I  in- 
terpret it,  consists  in  inducing  the  pupil  to  gain 
an  experience  of  his  own  which  will  enable  him  to 
regard  every  definition  as  the  result  of  his  own 
personal  observation  and  thought,  every  rule  as  a 
statement  of  the  method  by  which  he  has  done 
something,  and  every  new  word  as  only  a  labor- 
saving  device  for  the  expression  of  a  familiar  idea. 
This  result  is  accomplished  by  a  series  of  questions 
so  framed  as  to  lead  the  pupil  to  draw  such  con- 
clusions from  what  he  already  knows  as  will  con- 
stitute new  elements  of  knowledge  on  which  in 
turn  new  questions  may  be  based.  These  ques- 
tions at  the  beginning  of  each  subject  are  made 

1  See  Stocks,  Duties,  Taxes,  etc. 


Preface.  v 

simple,  and  require  the  use  of  small  numbers  only, 
so  that  the  mind  of  the  learner  may  be  occupied 
with  principles  rather  than  with  their  mechani- 
cal applications.  More  complicated  relations  and 
larger  numbers  are  gradually  introduced  until 
finally  the  pupil  finds  it  necessary  to  make  a 
record  of  some  of  his  work;  his  written  work, 
however,  is  always  intended  to  be  a  record  of  the 
results,  not  of  mechanical  processes,  but  of  mental 
operations. 

In  the  application  of  this  method  the  duty  of 
the  teacher  is  merely  to  provide  material  for  work 
and  to  see  that  it  is  done ;  the  duty  of  the  pupil 
is  to  work  and  to  discover.  In  teaching  we  are 
strongly  tempted  to  give  the  pupil  the  benefit  of 
our  own  experience,  hoping  thus  to  shield  him 
from  the  toilsome  process  of  gaining  for  himself 
that  experience  which  we  have  acquired  only 
through  hard  work,  but  if  we  call  to  mind  the  pro- 
cess of  learning  to  do  that  which  we  do  best  we 
shall  find  that  it  has  consisted,  not  in  learning  by 
heart  the  results  of  the  experience  of  others,  but 
in  individual,  manly  effort  applied  in  the  spirit  of 
investigation  and  discovery. 

Although  this  book  is  intended  as  a  continua- 
tion of  Warren  Colburn's  First  Lessons,  I  have 
drawn  from  the  First  Lessons,  with  permission, 
enough  matter  relating  to  Fractions  to  form  a 


vi  Preface. 

good  review  of  the  subject.  I  have  also  repeated 
here  Chapters  VI.  and  VII.  which  relate  to  writ- 
ten addition,  subtraction,  multiplication,  and  divi- 
sion ;  these  chapters  did  not  form  a  part  of  the 
original  edition  of  the  First  Lessons  but  were  pre- 
pared by  me  for  the  revised  edition  of  1884.  The 
matter  taken  from  the  First  Lessons  is  carefully 
indicated  wherever  it  occurs  so  that  it  may  be 
readily  omitted  by  pupils  who  are  already  suffi- 
ciently familiar  with  it. 

The  Second  Lessons,  while  complete  in  itself, 
can  (as  can  any  other  book  of  the  same  scope)  be 
used  to  the  best  advantage  by  pupils  who  have 
already  studied  Warren  Colburn's  First  Lessons. 

H.  N.  W. 
CAMBRIDGE,  Mass.,  Aug.  31, 1888. 


CONTENTS. 


SECTION  I. 

PAGE 

Notation,  Addition,  and  Subtraction    ....  1-30 

A.  Questions  on  Notation,  with  Explanations          .         .  1-6 

B.  Addition,  with  Practical  Illustrations         .         .         .  6-14 

C.  Subtraction,  with  Practical  Illustrations    .         .         .  14-21 

D.  United  States  Money 21-24 

E.  Tables  and  Questions  for  Practice     ....  25-30 

SECTION  II. 

Multiplication  and  Division 31-64 

A.  Multiplication:    Examples  and  Problems,  with  re- 

marks and  explanations          .....  31-45 

B.  Bills 46-51 

C.  Division:    Examples   and  Problems,  with  remarks 

and  explanations 51-62 

D.  Tables  and  Questions  for  Practice     ....  62-64 

SECTION  III. 

Fractions 65-99 

A.  Fractional  Notation  and  Fractional  Terms        .         .  65-68 

B.  Common  Denominator.     Problems.     Illustrations     .  68-83 

C.  Multiplication  of  Fractions 83-90 

D.  Division  of  Fractions 90-95 

E.  Miscellaneous  Questions 96-99 

SECTION  IV. 

Decimals  :  Introduction.  Notation  with  simple  il- 
lustrative examples  in  Addition,  Subtraction, 
Multiplication,  and  Division  ....  100-116 


viii  Contents. 


SECTION  V. 

Multiplication  of  Decimals 117-128 

A.  Multiplication  of  a  Decimal  by  a  Whole  Number. 
Examples  and  Problems,  with  remarks  and  expla- 
nations          117-122 

B.  Multiplication  of  a  Decimal  by  a  Decimal.     Ex- 

amples and  Problems,  with  remarks  and  expla- 
nations            122-128 

SECTION  VI. 

Division  of  Decimals 129-147 

A.  Division   by  a  Whole   Number.      Examples  and 

Problems,  with  remarks  and  explanations  .    12!M.'I7 

B.  Division  of  a  Decimal  by  a  Decimal.     Examples 

and  Problems,  with  remarks  and  explanations  .    137-143 

C.  Reduction   of   Common    Fractions    to    Decimals. 

Circulating  Decimals  .....    143-145 

D.  Miscellaneous  Examples         .....    145-147 

SECTION  VII. 

Percentage 148-205 

A.  Interest 148-167 

B.  Compound  Interest 167-171 

^  C.  Partial  Payments 172-176 

D.  Equation  of  Payments  .         .         .         .        .         .  176-179 

E.  Stocks 179-185 

F.  Taxes 185-188 

G.  Duties 189-193 

H.  Miscellaneous 194-205 

SECTION  VIII. 

Square  and  Solid  Measures 206-225 

A.  Square  Measure 206-215 

B.  Solid  Measure 215-222 

C.  Board  Measure 223-225 

SECTION  IX. 
Divisors,  Factors,  and  Multiples       ....  226-242 


Contents. 


IX 


SECTION  X. 
Cancellation  and  Analysis 


243-258 


APPENDIX. 
CHAPTER  I. 


Roman  Notation  . 


259-262 


CHAPTER  II. 

The  Metric  System  of  Measures 

A.  Linear  Measure      .... 

B.  Square  Measure      .... 

C.  Solid  Measure          .... 

D.  Capacity  Measure  .... 

E.  Weight  Measure     .... 

F.  Miscellaneous          .... 


262-275 

268-267 

267-270 

270-271 

271-272 

272 

273-275 


CHAPTER  III. 


Arithmetical  Tables     . 
United  States  Money 
English  Money 
French  Money 
German  Money 
Length   . 

Surveyor's  Measure 
Surface  . 
Solidity  . 
Liquid  Measure 
Dry  Measure  . 
Avoirdupois  Weight 
Troy  Weight 
Apothecaries'  Weight 
Apothecaries'  Measure 
Time       . 
Miscellaneous 


275-282 

275 

275 

275 

275 

276 
,  276 

276 
,  277 

277 

277 

277 
,  278 
,  278 
,  278 
,  279 
,  279 


Contents. 

The  Metric  System.  Linear  Measure  .  .  .  280 

Square  Measure 280 

Cubic  Measure 281 

Capacity  Measure 281 

Weights 281 

Compound  Interest  Table 282 


SECTION  L 

NOTATION,  ADDITION,  AND  SUB- 
TRACTION. 

The  portions  of  this  section  that  are  preceded  by  a  star  (*)  have 
been  drawn  from  Section  VI.  of  the  new  edition  of  Colburn's  First 
Lessons,  and  may  at  the  discretion  of  the  teacher  be  omitted  by 
pupils  who  have  studied  them  before. 

A.   Questions  on  Notation,  with  Explanations. 

*1.  What  number  is  represented  by  36  ? 

Answer:  Thirty-six. 

What  does  the  3  stand  for  ?  the  6  ? 

Answer :  36  =  3  tens  +  6  units  ;  the  3,  then, 
stands  for  the  whole  number  of  tens  in  thirty-six, 
and  the  6  for  the  units  left  over. 

*2.  What  do  the  figures  in  42  represent?  in 
69?  in  58?  in  17?  in  23? 

*3.  What  does  the  right-hand  figure  of  a  num- 
ber denote  ?  what  does  the  figure  to  the  left  of 
the  right-hand  figure  denote  ? 

Since  the  right-hand  figure  denotes  units,  and 
the  figure  to  the  left  of  ifc  denotes  tens,  we  say 
that  the  first  figure  from  the  right  occupies  the 
units9  place,  and  that  the  second  figure  from  the 
right  occupies  the  tens9  place. 


2         Notation,  Addition,  and  Subtraction.   [§  1. 

*4.  What  is  the  number  that  has  8  in  the  ^  . 
units'  place  and  6  in  the  tens'  place?  (An-  jj  g 
swer :  Sixty-eight.)  9  in  the  units'  place  and  6  8 
7  in  the  tens'  place  ?  5  in  the  tens'  place  and  7  9 
2  in  the  units'  place?  4  in  the  tens'  place  5  2 
and  0  in  the  units'  place  ?  40 

*5.  What  number  is  represented  by  857  ? 

Answer:  Eight  hundred  and  fifty-seven. 
What  do  the  different  figures  stand  for? 
Answer  :   857  =  8  hundred  +  5  tens  +  7  units ; 
the  8,  then,  stands  for  the  whole  number  of  hun- 
dreds, the  5  for  the  whole  number  of  tens  left 
over,  and  the  7  for  the  units  remaining  after  tak- 
ing away  both  the  hundreds  and  the  tens. 

*6.  What  do  the  different  figures  represent  in 
614?  in  891?  in  412?  in  508?  in  320? 

*7.  What  does  the  figure  to  the  left  of  the 
tens'  place  denote  ? 

*8.  Which  place,  counting  from  the  right,  is 
the  hundreds'  place  ? 

*9.  What  is  the  number  that  has  6  in     rf 
the  units'  place,  8  in  the  tens'  place,  and     I   ^   «• 
4  in  the  hundreds'  place  ?  (Answer :  Four     £  Jj  j| 
hundred   and  eighty-six.)    9  in  the  tens'     486 
place,  7  in  the  hundreds'  place,  and  2  in     792 
the  units'  place  ?  6  in  the  units'  place,  0     806 
in  the  tens'  place,  and  8  in  the  hundreds'     600 
place  ?  6  in  the  hundreds'  place,  0  in  the 
;  ens'  place,  and  0  in  the  units'  place  ? 

*1O.  The  number  320  stands  for  32  tens,  but  it 
is  often  convenient  to  call  10  tens  a  "  hundred," 


A.]  Questions  on  Notation,  with  Explanations.  3 

so  that  32  tens  is  3  hundred  and  2  tens,  or  three 
hundred  and  twenty.  Similarly  we  call,  for  con- 
venience, 10  hundred  (1000)  a  "  thousand  "  ;  20 
hundred  (2000)  2  "  thousand  "  ;  30 
hundred  (3000)  3  "  thousand "  ;  | 

3600,  then,  stands  for  36  hundred, 
or  3  thousand  and  6  hundred; 
13200  for  132  hundred,  or  13  thou- 
sand  and  2  hundred;  168362  for 
168  thousand  3  hundred  and  62. 

Write  in  words  each  of  the 
numbers  in  the  adjacent  column. 
First  number  :  Three  hundred  and  982742 
twenty-eight  thousand  nine  hundred  and  sixty - 
eight. 

*11.  What  figures  stand  for  thousands  in  each 
of  the  above  numbers  ?  what  figure  stands  for  hun- 
dreds ?  for  tens  ?  for  units  ? 

*12.  Counting  from  the  right,  what  places  do 
the  figures  that  represent  thousands  occupy  ? 

In  order  to  make  it  easier  to  read  a  number,  it 
is  usual  to  separate  by  a  comma  the  figures  that 
denote  thousands  from  the  figure  that  denotes  hun- 
dreds ;  thus,  in  198,642  we  place  a  comma  between 
the  8  and  the  6,  and  read  198  thousand  6  hundred 
and  42. 

*13.  Read :  986,432  ;  16,768  ;  270,432  ;  7,487  ; 
100,708;  908,550;  2,008;  68,052;  111,684; 
777,777. 

*14.  Express  by  figures  the  numbers  : 

Seventy-seven  thousand  six  hundred  and  eighty- 
four. 


4         Notation,  Addition,  and  Subtraction.   [§  1. 

Thirty-four  thousand  and  fifty-six. 

Four  hundred  and  seventeen  thousand  and  six 
hundred. 

Seven  hundred  and  fourteen  thousand  and  six 
hundred. 

Six  hundred  thousand  and  two. 

Three  hundred  and  ninety-one  thousand  six  hun- 
dred and  twelve. 

Fifty-seven  thousand  one  hundred  and  nine. 

Nine  hundred  thousand  four  hundred  and  three. 

*15.  1,000,000,  or  1  thousand  thousand,  is  us- 
ually called  a  "  million."     16,000,000,  or  16  thou- 
sand thousand,  may  be  called,  then, 
16    million  ;     186,000,000,   or   186       g      |   | 
thousand   thousand,    may  be   called       |       |   n«..s 
186  million.  J| j^AS 

186  million,  186,000,000. 

432  thousand,  432,000. 

186  million  432  thousand  and  792,    186,432,792. 

799  million  and  684,  799,000,684. 

896  million  and  479  thousand,          896,479,000. 

Counting  from  the  right,  what  places  do  the 
figures  that  represent  millions  occupy? 

*16.  What  do  we  call  the  first  place  from  the 
right  ?  the  second  place  ?  the  third  place  ?  the 
fourth,  fifth,  and  sixth  places  ?  the  seventh,  eighth, 
and  ninth  places  ? 

*17.  Read  the  numbers  : 
982,461,007        111,111,111          2,760,286 
698,231,770        232,008,674        16,888,888 
842,000,689          11,400,800      769,000,001. 


A.]  Questions  on  Notation,  with  Explanations.  5 

*  1 8.  Express  by  figures  the  numbers  : 

Six  million  seventy-five  thousand  and  four. 

Three  hundred  and  six  million  and  forty  thou- 
sand. 

Five  million  six  hundred  and  seventeen  thousand 
and  forty-three. 

Four  hundred  and  sixty  million  and  twenty- 
seven. 

Seventy-three  million  forty-one  thousand  eight 
hundred  and  nineteen. 

*19.  Express  by  figures  the  numbers  : 

Nine  hundred  million  one  hundred  and  seven. 

Eighty-six  million  and  seven  hundred. 

Thirteen  hundred  and  eighty-six  million  four 
hundred  and  ninety-seven  thousand  three  hundred 
and  sixty-two. 

20.  Light  travels  one  hundred  and  eighty-six 
thousand  three  hundred  and  twenty-four  miles  in  a 
second  ;  express  this  number  of  miles  by  figures. 

21.  Sound  travels  eleven  hundred  and  twenty 
feet  in  a  second  ;  express  this  number  of  feet  by 
figures. 

22.  The   distance   around    the    earth    at   the 
equator  is  twenty-four  thousand  eight  hundred  and 
ninety-nine  miles ;   express  this  number  of  miles 
by  figures. 

23.  The  distance  of  the  earth  from  the  sun  is 
ninety-five  million  miles  :  express  this  number  of 
miles  by  figures. 

24.  The  distance  of  the  earth  from  the  moon  is 
two  hundred  thirty-eight  thousand  eight  hundred 


6         Notation,  Addition,  and  Subtraction.  [§  1. 

and   forty-eight   miles;    express   this   number   of 
miles  by  figures. 

25.  During  the  civil  war  of  1861  to  1865  two 
million  eight  hundred  and  fifty-nine  thousand  one 
hundred  and  thirty-two  men  were  supplied  to  the 
Union  army;  express  this  number  of  men  by 
figures. 

B.  Addition,  with  Practical  Illustrations. 

*1.  How  many  complete  rows  of  ten  each  can 
you   make   with    32    counters  ?  how  many 
counters  will  be  left  over  ? 

Answer :   3  complete  rows  of  ten,  and     •  •  • 
there  will  be  two  extra  counters.  •  •  • 

*2.  How  many  complete  rows  of  ten  each 
can  you  make  with  24  counters  ?  how  many     •  •  • 
counters  will  be  left  over  ? 

*3.  How   many   complete   rows   of  ten 
counters  each  can   you  make  with  49  counters? 
83  counters?    61  counters?    17  counters?     How 
many  extra  counters  will  there  be  in  each  of  these 
cases  ? 

*4.  How  many  counters  must  you  have  in  order 
to  make  5  complete  rows  and  have  3  count- 
ers over  ?  7  rows  and  2  counters  over  ? 

*5.  How  many  complete  rows  of  ten  each 
can  you  make  with  32  counters  and  24  •• 
counters,  and  how  many  counters  will  be  ** 
left  over  ?  •  • 

Answer :   It  is  easy  to  see,  by  looking  at     J  J 
the  diagrams  given  above,  that  with  32  count- 


B.j     Addition,  with  Practical  Illustrations.       7 

ers  we  should  have  3  rows  and  2  extra  counters  ; 
with  24  counters  we  should  have  2  rows  and  4  extra 
counters  ;  with  32  and  24  counters  we  should  then 
have  3  +  2  rows  and  2  +  4  extra  counters,  or  5  rows 
and  6  counters. 

*6.  How  many  complete  rows  of  ten  counters 
each  can  you  make  with  42  counters  +  53  counters? 
with  13  counters  +  22  counters  ?  How  many  count- 
ers will  be  left  over  in  each  case  ? 

*7.  Mr.  Smith  paid  23  cents  for  a  piece  of 
cheese,  and  36  cents  for  some  butter  ;  how  much 
did  he  pay  in  all? 

*8.  Mrs.  Jones  paid  64  cents  for  veal,  and  23 
cents  for  vegetables  ;  how  much  did  she  pay  for 
her  dinner  ? 

Let  us  solve  this  problem  by  the  aid  of  counters. 


JJJJJJ*  (  64  counters  =  6  rows  of  tens +  4  count- 


|  23  counters  =  2  rows  of  tens +  3  count- 
ers. 


In  all  there  are  8  rows  of  tens  and  7  extra 
counters,  or  87  counters  ;  therefore,  64  +  23  -  87, 
and  Mrs.  Jones  paid  87  cents  for  her  dinner. 


8         Notation,  Addition,  and  Subtraction.   [§  1. 
*9.  How  many  are  334-26  ?  44  +  16  ? 


}•    =  33 


=  26 


=  44 


=  16 


5  tens  +  9  =  59.         5  tens  +  10  =  6  tens  =  60. 

*10.  Add   35    41     97    23     84    16     80    41 
to       43JL8_22561123^228 

Answers:  78"591197995399269. 
*11.  How  many  are  27  +  35? 


__  07  In  the  two  incomplete  rows  there 
are  7  +  5  or  12  counters,  which 
make  one  row,  with  2  counters 
left  over :  all  together,  then,  there 
are  (1  +  5)  6  rows  of  tens +  2  count- 
ers, or  62  counters. 

=  35         Therefore,  27  +  35  =  62. 


€•• 


B.]     Addition,  with  Practical  Illustrations.       9 
*12.  How  many  are  36  +  28  ? 


}>  =36 

The  extra  counters  make  1  row 
and  4  counters :  all  together,  then, 
there  are  6  rows  and  4  counters. 

Therefore,  36  +  28  =  64. 

h  =28 


*13.  Add    23    24    16     35     83    44    76    31 
to        18     67     17    48     29     39    16    49 

Answers:   41    ~91     33     83  112     83     92     80. 

WITHOUT   COUNTERS. 

*14.  How  many  are  74  +  68  ? 
74=   7  tens+   4  units. 
68=   6  tens+    8  units. 


74  +  68  =  13  tens  + 12  units 

=  14  tens  +   2  units  =  142. 
Or,  more  briefly, 

74         8  and  4  are  12,  or  1  ten +  2  over  ;  we  set 

68     down  the  2  and  save  the  1  ten.     1  ten  (the 

142     one  that  was  saved)  and  6  are  7  and  7  are 

14  tens ;  we  set  down  the  14  to  the  left  of 

the  2,  and  have  142  for  an  answer. 

*15.  Add    56         8  and  6  are  14  ;  we  set  down 

to         28     the  4  and  save  the  1.     1  (the 

Answer :      84     1  that  was  saved)  and  2  are  3, 


10       Notation,  Addition,  and  Subtraction.  [§  1. 

and  5  are  8  ;  we  set  down  the  8  to  the  left  of  the 
4,  and  get  84  for  our  answer. 


*16.  Add 
to 

Answers  : 

25 
65 

89 
98 

57 
75 

84 
29 

76 
56 

90 

187 

132 

113 

132. 

*17.  How  many  are  269  +  328? 

269  =  2  hundred +  6  tens +9  units. 
328  =  3  hundred  +  2  tens  +  8  units. 

269  +  328  -5  hundred  +  8  tens-f   I  17  units 

(  or  1  ten  4- 7  units 

=  5  hundred  +  9  tens  +  7  units 

-597. 
*18.  How  many  are  684  +  767? 

684=   6  hundred  +    8  tens  +   4  units. 
767=   7  hundred  +   6  tens  +   7  units. 

684  +  767  =  13  hundred +  14  tens +  11  units 
=  13  hundred  + 15  tens  +   1  unit 
=  14  hundred  +   5  tens  +   1  unit 
=  1451. 

*19.  Add  793         184        Write  out  your 

to  848         678  work. 

*20.  Add    768         7  and  8  are  15 ;  we  set  down 

to        857     the  5  and  save  the  1.     1  and 

1625     ^  are  6,  and  6  are  12  ;  we  set 

down  the   2  and  save  the  1. 

1  and  8  are  9,  and  7  are  16  ;  we  set  down  the  16, 

and  have  for  an  answer  1625. 

*21.  Add  689          972          439          139 

to  763          684          698          984 

Answers:     1452        1656        1137        1123. 


B.]     Addition,  ivith  Practical  Illustrations.      11 

*22.  Add  858          900          237          642 

to  686          768          508          899 

*23.  Add    667         8  and  8  are  16,  and  6  are 

276     22,  and  7  are  29  ;  we  set  down 

108     the  9  and  save  the  2.     2  (the 

188     2  that  was  saved)  and  8  are 

1239     10,  and  7  are  17,  and  6  are 

23  ;  we  set  down  the  3  and 

save  the  2.     2  and  1  are  3,  and  1  are  4,  and  2  are 

6,  and  6  are  12  ;  we  set  down  the  12.     We  have 

for  an  answer  1239. 

*24.  Add 


284 

326 

522 

683 

355 

719 

799 

793 

123 

268 

655 

573 

618 

400 

321 

498 

1380        1713        2297        2547 


687 

322 

623 

60 

864 

921 

888 

798 

730 

777 

760 

666  • 

231 

815 

298 

208 

419 

23 

402 

476 

*25.  America  was  discovered  by  Columbus  in 
1492,  and  the  War  for  Independence  began  283 
years  afterwards  ;  in  what  year  did  this  war  begin  ? 

*26.  In  July,  1776,  the  Declaration  of  Inde- 
pendence was  made  ;  87  years  afterwards  the  Bat- 
tle of  Gettysburg  was  fought  •.  find  the  date  of  the 
Battle  of  Gettysburg. 

*27.  A  butcher  bought  3  oxen  :  the  first  weighed 


12       Notation,  Addition,  and  Subtraction.   [§  1. 

1214  pounds,  the  second  1406  pounds,  and  the 
third  1384  pounds ;  how  much  live-meat  had  he 
in  all? 

28.  How  many  feet  of  fence  will  be  required  to 
surround  a  house-lot,  the  four  sides  of  which  are 
126  feet,  233  feet,  126  feet,  and  233  feet  ? 

29.  A  man  paid  250  dollars  for  a  carriage,  225 
dollars  for  a  horse,  3  dollars  for  a  whip,  and  6 
dollars  for  a  robe  ;  what  did  they  all  cost  ? 

30.  How  many   public    school    children    were 
there  in  New  England  in  1885  if  in  Maine  there 
were  145,317,  in  New  Hampshire  64,219,  in  Ver- 
mont 71,667,  in  Massachusetts  349,617,  in  Rhode 
Island  47,882,  and  in  Connecticut  125,539  ? 

31.  The  following  is  a  summary  of  the  students 
in  Harvard  University  in  October,  1887  :  In  the 
Undergraduate  Department  1138,  in  the  Divinity 
School  16,  in  the  Law  School  215,  in  the  Scientific 
School  20,   in   the   Medical   School   263,  in   the 
Dental  School  32,  in  the  Bussey  Institution  7,  in 
the    School   of   Veterinary   Medicine   26,   in   the 
Graduate   Department   96 ;    how   many   students 
were  there  in  all  ? 

32.  The  following  is  a  summary  of  the  students 
in  Yale  University  in  October,  1887  :  In  the  Un- 
dergraduate   Department    614,    in    the    Divinity 
School  117,  in  the  Law  School  94,  in  the  Scien- 
tific School  291,  in  the  Medical  School  26,  in  the 
School  of  Fine  Arts  58,  in  the  Graduate  Depart- 
ment 69  ;  how  many  students  were  there  in  all  ? 

33.  George  Washington  was  born  in  the  year 


B.]     Addition,  with  Practical  Illustrations.     13 

1732  ;  he  was  elected  President  when  57  years  old, 
and  died  10  years  afterwards ;  in  what  year  did 
he  die  ? 

34.  The  difference  of  two  numbers  is  960,843, 
and  the  smaller  number  is  229,317  ;  what  is  the 
larger  number  ? 

35.  The  distance  from  Washington  to  Baltimore 
being  38  miles,  thence  to  Philadelphia  99  miles, 
thence  to  New  York  90  miles,  thence  to  Worcester 
175  miles,  thence  to  Boston  44  miles ;  how  far  is 
Boston  from  Washington  ? 

36.  How  many  days  are  there  in  a  leap  year, 
there  being  7  months  of  31  days  each,  one  month 
of  29  days,  and  4  months  of  30  days  each. 

37.  How  many  strokes  does  a  common  clock 
strike  in  24  hours? 

38.  When  you  go  from  Boston  to  San  Francisco, 
if  you  take  the  Hoosac   Tunnel  Route   through 
Buffalo  and  Detroit  to  Chicago,  the  Chicago,  Bur- 
lington, and  Quincy  road  from  Chicago  to  Denver, 
the  Denver  and  Rio  Grande  road  from  Denver  to 
Ogden  (Utah),  and  the  Central  Pacific  road  from 
Ogden  to  San  Francisco,  you  will  travel  150  miles 
in  Massachusetts,   336   miles  in  New  York,   236 
miles  in  Canada,  222  miles  in  Michigan,  41  miles  in 
Indiana,  227  miles  in  Illinois,  276  miles  in  Iowa, 
371    miles  in   Nebraska,  623  miles   in  Colorado, 
473  miles  in  Utah,  461  miles  in  Nevada,  and  282 
miles   in   California ;    how  many  miles   will  you 
travel  in  all  ? 

39.  When  you  go  from  Boston  to  the  City  of 


14     Notation,  Addition,  and  Subtraction.     [§  1. 

Mexico,  if  you  take  the  Hoosac  Tunnel  Route  to 
Chicago,  as  indicated  in  the  last  example,  the 
Chicago,  Burlington,  and  Quincy  road  from  Chi- 
cago to  Kansas  City,  the  Atchison,  Topeka,  and 
Sante  Fe  road  to  El  Paso,  and  the  Mexican  Cen- 
tral road  from  El  Paso  to  the  City  of  Mexico, 
you  will  travel,  after  leaving  Chicago,  263  miles 
in  Illinois,  224  miles  in  Missouri,  485  miles  in 
Kansas,  190  miles  in  Colorado,  498  miles  in  New 
Mexico,  and  1225  miles  in  Mexico ;  what  will  be 
the  length  of  your  entire  journey  ? 


C.  Subtraction,  with  Practical  Illustrations. 

*1.  Take  away  8  counters  from  35  counters  and 
how  many  will  remain  ? 


^  =35  counters  =3  rows +  5  counters. 


To  take  away  8  counters  we  first  take  away  the 
5  extra  counters,  and  then  going  to  the  next  row 
we  keep  on  taking  counters  away  until  in  all  we 
have  taken  away  8  ;  counting  those  that  remain, 
we  find  that  there  are  2  rows  +  7  counters  or  27 
counters. 

35  less  8  are  how  many  ? 


C.]  Subtraction,  with  Practical  Illustrations.  15 

*2.  Take  away  18  counters  from  35  counters 
and  how  many  will  remain  ? 

The  35  counters  are  shown  in  the  last  question. 

18  counters  =  1  row +  8  counters. 

After  taking  away  first  the  8  counters,  and  then 
the  1  row,  we  find  by  counting  that  17  counters 
remain. 

35  less  18  are  how  many  ? 

*3.  From  82  71  95  76 

take  23          45  37          59 

Answers:  59  26  58  17 

Illustrate  by  counters. 

WITHOUT   COUNTERS. 

*4.  How  many  are  48  less  32  ? 
48  =  4  tens +  8  units. 
32  =  3  tens +  2  units. 


48  -  32  =JL  ten  +  6  units  =  16.  Answer. 
*5.  From  68  99          84  74 

take  42  76          jtt  32 

Answers:  26  23  53  42 

*6.  How  many  are  53  less  18  ? 

53  =  5  tens +3  units. 

18  =  1  ten  +8  units. 


We  cannot  take  8  units  from  3  units  ;  we  there- 
fore take  one  of  the  5  tens  and  add  it  to  the  3 
units. 

53  =  4  tens +  13  units. 

18  =  1  ten  +    8  units. 


63  -  18  =  3  tens  +   5  units  =  35.  Answer. 


16     Notation,  Addition,  and  Subtraction.     [§  1. 

*7.  How  many  are  97  less  68  ? 
97  =  8  tens +  17  units. 
68  =  6  tens+    8  units. 


97  -  68  =  2  tens  4-    9  units  =  29.  Answer. 
*8.  From         77  97  73  36 

take          38         ^9          58          28 

Answers:  39  48  15  8 

*9.  A  farmer  raised  89  bushels  of  potatoes  ;  he 
kept  18  bushels  for  his  family,  and  sold  the  rest : 
how  many  bushels  did  he  sell  ? 

*10.  A  man  who  owed  a  bill  of  96  dollars,  paid 
20  dollars  at  one  time,  and  19  dollars  at  another ; 
how  much  did  he  then  owe  ? 

*11.  Mr.  Jackson  raised  78  dollars'  worth  of 
hay  above  what  he  needed  to  feed  to  his  stock  ;  he 
sold  this  hay  to  the  grocer  and  received  in  part 
pay  29  dollars'  worth  of  flour :  how  much  money 
should  he  receive  ? 

*12.  Mr.  Weston  started  to  walk  from  Ports- 
mouth to  Boston,  a  distance  of  56  miles :  he 
walked  34  miles  on  the  first  day ;  how  many  miles 
had  he  left  to  go  on  the  second  day  ? 

*13.  Mr.  Fowler,  the  mason,  brought  73  bricks 
in  his  wheelbarrow  to  finish  the  arch  he  was  build- 
ing :  he  used  only  57  of  them  ;  how  many  had  he 
left? 

*14.  Mr.  Knapp's  parlor  is  97  inches  high  :  the 
wainscoting  above  which  the  room  is  to  be  papered 
is  33  inches  high.  Into  what  lengths  must  the 
paper  be  cut  ? 


C.]  Subtraction,  with  Practical  Illustrations.  17 

*15.  Mr.  Lanman  sent  67  books  to  be  bound: 
the  binder  made  mistakes  in  the  lettering  of  19  of 
them  ;  how  many  were  lettered  correctly  ? 
*16.  683  less  421  are  how  many? 

From     683  =  6  hundred  +  8  tens  +  3  units, 
take       421  =  4  hundred  +  2  tens  + 1  unit. 

683-421  =  2  hundred +  6  tens +  2  units 

=  262.  Answer. 

*17.  a.  From     896         764         972         541 
take       784        451         630         321 

Answers:       112         JS13         342       ~220 

6.  From         468         532         769        497 
take  351        412        347         182 

*18.  a.  How  many  are  684  less  296  ? 

From  684  =  6  hundred +  8  tens +  4  units, 
take    296  =  2  hundred  +  9  tens  +  6  units. 
We  cannot  take  6  units  from  4  units  ;  we  there- 
fore take  one  of  the  8  tens  and  add  it  to  the  4 
units,  so  that  the  problem  will  read : 

From  684  =  6  hundred +  7  tens +  14  units, 
take    296  =  2  hundred +9  tens-h   6  units. 
We  cannot  take  9  tens  from  7  tens  ;  we  there- 
fore take  one  of  the  6  hundred  and  add  it  to  the 
7  tens  ;  and  our  problem  is  now  as  follows  : 

From     684  =  5  hundred  + 17  tens  + 14  units, 
take       296  =  2  hundred  +    9  tens  +   6  units. 

684-296  =  3  hundred  +   8  tens  +   8  units 

=  388.  Answer, 
b.  Show  that  762  less  484  are  278. 


18     Notation,  Addition,  and  Subtraction.     [§  1. 


>19.  From 
take 

Answers  : 

811 
422 

623 
237 

762 
478 

436 
184 

389 

386 

284 

252. 

20.  From 
take 

431 

292 

973 

584 

763 
184 

842 
754 

*21.  From  974  take  783. 

We  may  state  briefly  what  we  do  thus  :  3  from 
4  gives  1 ;  we  set  down  the  1  in  the  units'  974 
column.  One  of  the  9  hundred  added  to  783 
the  7  tens  gives  17  tens,  and  8  tens  from  191 
17  tens  are  9  tens ;  we  set  down  the  9  in  the  tens' 

column.  7  hundred  from  the  remaining  8  hun- 
dred is  1  hundred ;  we  set  down  the  1  in  the  hun- 
dreds' column. 

We  have,  then,  191  for  an  answer. 

*22.  From         864         726         419  521 

take  471        233        126  197 

Answers:        393        493         293  324. 

*23.  From  921  take  246. 

One  of  the  2  tens  added  to  the  1  unit  gives 
11  units,  and  6  from  11  are  5  ;  we  set  921 
down  the  5  in  the  units'  column.  One  of 
the  9  hundred  added  to  the  remaining  1  675 
ten  gives  11  tens,  and  4  tens  from  11  tens  are  7 
tens ;  we  set  down  the  7  in  the  tens'  column.  2 
hundred  from  the  remaining  8  hundred  are  6 
hundred ;  we  set  down  the  6  in  the  hundreds' 
column. 

We  have,  then,  675  for  an  answer. 


C.]  Subtraction,  with  Practical  Illustrations.  19 

*24.  From          684         555         911         722 
take  296         166         223         333 

Answers:       ~388        ~389       ~688         389. 
*25.  From  842  take  468. 
We  may  state  our  work  very  briefly  thus : 
8  from  12  are  4  ;  842 

6  from  13  are  7  ;  468 

4  from    7  are  3.  374 

*26.   From    633  6  from  13  are  7 

take      216  l  from    2  is     1 

Answer :    417  2  from    6  are  4. 

*27.  From    846  4  from    6  are  2 

take       674  7  from  14  are  7 

Answer:    172  6  from    7  is     1. 

*28.  From     811  2  from  11  are  9 

take       322  2  from  10  are  8 

Answer :     489  3  from    7  are  4. 

*29.  From    600  8  from  10  are  2 

take      268  6  from    9  are  3 

Answer :    332  2  from    5  are  3. 

*3O.  America  was  discovered  by  Columbus  in 
the  year  1492 ;  how  many  years  ago  was  that  ? 

*31.  The  Pilgrims  landed  at  Plymouth  in  1620  ; 
how  many  years  ago  was  that?  how  many  years 
after  the  discovery  of  America  by  Columbus  ? 

*32.  Harvard  College  was  founded  in  the  year 
1636  ;  how  many  years  ago  was  that  ? 

33.  In  1880  London  had  3,832,441  inhabitants, 
and  New  York  only  1,206,590  ;  how  man^  more 
inhabitants  had  London  than  New  York  ? 


20     Notation,  Addition^  and  Subtraction.     [§  1. 

34.  From  6942  7123         12111 
take              3369          6876          8467 

35.  If  I  buy  a  house  for  5875  dollars  and  sell 
it  for  7100  dollars,  do  I  gain  or  lose,  and  how 
much? 

36.  The  population  of  the  United  States  in  1840 
was  17,069,453  ;  in  1880,  49,369,595  ;  what  was 
the  increase  of  population  during  these  forty  years  ? 

37.  Subtract  1907  from  11442  until  nothing  re- 
mains. 

38.  Benjamin  Franklin  died  in  1790,  aged  84 
years  ;  in  what  year  was  he  born  ? 

39.  If  William  is  23  years  old,  in  what  year  was 
he  born  ? 

40.  From  the  sum  of  4936  and  7208  take  the 
sum  of  1137,  2065,  and  6820. 

41.  The   distance    from   Boston  to  Albany  by 
railroad  is  200  miles.     Suppose  one  locomotive  to 
have  gone  68  miles  from  Boston  towards  Albany, 
and  another  95  miles  from  Albany  towards  Boston ; 
how  far  are  they  apart  ? 

42.  The  larger  of  two  numbers  is  987,564,321, 
and  their  difference  is  14,097,738  ;   what  is  the 
smaller  number  ? 

43.  The  larger  of  two  numbers  is  842,260,084, 
and  their  difference  is  179,742,986;   what  is  the 
smaller  number? 

44.  In  1870  there  were  in  the  United  States 
19,493,565  males  and   19,064,806  females;  how 
many  more  males  were  there  than  females  ? 


D.]  United  States  Money.  21 

45.  In  1880  there  were  in  the  United  States 
25,518,820  males  and   24,636,963   females;  how 
many  more  males  were  there  than  females  ? 

46.  From  the  figures  given  in  the  last  two  ex- 
amples find  how  many  more  males  there  were  in 
the  United   States   in    1880   than  in  1870;  how 
many  more  females  in  1880  than  in  1870. 


D.    United  States  Money. 
10  mills  =  1  cent. 
10  cents  =  1  dime. 
10  dimes  or  100  cents  =1  dollar. 

*1.  The  sign  $  is  often  used  to  stand  for  dollars ; 
thus,  6  dollars  is  usually  written  $6. 

Eead  $12  ;  116  ;  124  ;  1196. 

*2.  Write  48  dollars ;  96  dollars  ;  132  dollars  ; 
2137  dollars ;  using  the  dollar  mark,  $,  in  each  case. 

*3.  A  man  bought  a  sail-boat  for  $325,  and  paid 
$88  for  a  new  set  of  sails,  and  $32  for  having  the 
boat  painted  ;  how  much  did  the  whole  cost  ? 

*4.  Mr.  Kresus  had  $1000  in  the  Lydian  Five- 
Cents  Savings  Bank,  but  he  drew  out  $432  to  pay 
for  a  lot  of  land  ;  how  much  was  there  left  in  the 
bank? 

*5.  A  dealer  bought  some  flour  for  $642,  and 
sold  it  at  a  gain  of  $97  ;  what  did  he  get  for  his 
flour? 

*6.  A  man  bought  a  lot  of  land  for  $672,  and 
paid  $167  for  having  it  graded  :  he  then  sold  the 


22     Notation,  Addition,  and  Subtraction.     [§  1. 

land  at  a  profit  of  $125  ;  how  much  did  he  get  for 
the  land  ? 

*7.  Write  37  dollars  and  24  cents. 
In  such  a  case  as  this  it  is  customary  to  write 
the  number  denoting  cents  after  the  number  denot- 
ing dollars  and  to  separate  the  two  numbers  by  a 
period,  thus :  137.24. 

In  the  same  way  we  write 

48  dollars  and  79  cents,  $48.79 

22  doUars  and  53  cents,  $22.53 

22  dollars  and  43  cents,  $22.43 

22  dollars  and  33  cents,  $22.33 

22  dollars  and  23  cents,  $22.23 

22  dollars  and  13  cents,  $22.13 

22  dollars  and  03  cents,  $22.03. 

Notice  that  when  the  number  of  cents  is  less 

than  10,  we  fill  out  the  empty  tens'  place  by  a 

zero.     Thus   we   do  not  write   22  dollars  and  3 

cents  $22.3,  but  $22.03. 

*8.  Read  $49.13 ;  $117.88 ;  $109.76  ;  $104.70 ; 
$99.04;  $98.10;  $92.07;  $100.01;  $417.62; 
$3189.25 ;  $4281.50. 

Notice  carefully  that  $625  stands  for  625  dollars, 
but  that  $6.25  stands  for  6  dollars  and  25  cents. 

*9.  Write  in  figures  : 
Four  hundred  and  seven  dollars. 
Fifty-six  dollars  and  thirty-seven  cents. 
Two  hundred  and  forty-three  dollars  and  five^cents. 
Five  thousand  six  hundred  and  forty  dollars  and 

nine  cents. 

Eight   thousand   seven  hundred  and  eighty-three 
dollars. 


D.]  United  States  Money.  23 

Eighty-seven  dollars  and  eighty-three  cents. 
Six  hundred  dollars  and  eight  cents. 
*1O.  85  cents  may  be  written  $0.85. 
Read  10.33;  $0.07;  $0.69;  $0.38;  $0.02. 
*1 1.  Write  in  the  same  way  48  cents  ;  83  cents  ; 
78  cents ;  50  cents  ;  4  cents. 

*12.  How  many  cents  are  there  in  $8.32  ? 

Answer :  832  cents. 

*13.  How  many  cents  are  there  in  $6.19?  in 
$41.31  ?  in  $64.09?  in  $3142.27  ? 
*14.  Show  that 

183  cents  are  equivalent  to  $  1.83. 
192  cents  are  equivalent  to  $  1.92. 
7189  cents  are  equivalent  to  $71.89. 
3787  cents  are  equivalent  to  $37.87. 
3601  cents  are  equivalent  to  $36.01. 
*15.  Add  $37.13  to  $43.81.     Answer :  $80.94. 
In  doing  such  examples  as  this,  write  the  num- 
bers in  a  column,  taking  care  to  write  cents  under 
cents  and  dollars  under  dollars  ;  the  numbers  may 
then  be  added  as  if  the  periods  were  not  there. 


*16.  Add 

$38.29 
$26.18 

$47.82 
$32.51 

$117.98 
$  21.08 

$64.47 

$80.33 

$139.06 

$316.47 

$283.32 

$4132.07 

$   8.17 

$397.09 
$  71.93 

$780.98 
$  0.18 
$  31.50 

*17.  Mrs.  Wentworth  paid  $128  for  a  chamber- 
set,  $46  for  mattresses  and  bed  linen,  $48.50  for 
a  carpet,  $24  for  curtains,  and  $32.38  for  other 


24     Notation,  Addition,  and  Subtraction.     [§  1. 

articles  needed  in  her  spare  chamber ;  what  was 
the  cost  of  furnishing  the  room? 

*18.  A  New  England  farmer  wished  to  move 
out  West,  so  he  sold  his  farm  and  his  stock  and 
with  the  proceeds  went  to  Kansas.  He  sold  his 
two  horses  for  $246,  his  oxen  for  $98,  his  sheep 
for  $136,  his  pigs  for  $24.50,  and  his  poultry  for 
$16.48  ;  how  much  did  he  get  in  all  from  the  sale 
of  his  stock  ? 

*19.  On  the  morning  of  the  8th  of  May  Mr. 
Brown  had  only  $186.14  in  the  bank,  but  in  the 
course  of  the  day  he  deposited  $112  in  bills,  and 
checks  amounting  to  $347.32 ;  how  much  had  he 
then  in  the  bank  ? 


*20.  Add 


A 

B 

C 

14268.07 

$138.06 

$6842.16 

$6842.16 

$291.98 

$6843.27 

$7999.99 

$236.72 

$7234.82 

$2316.11 

$972.36 

$6124.86 

$7000.98 

$842.16 

$5555.55 

$1119.72 

$777.77 

$3232.39 

$2942.15 

$666.66 

$9998.99 

$3715.35 

$545.53 

$6849.76 

$1872.89 

$379.23 

$2332.29 

$6792.91 

$192.98 

$6974.84 

21.  Take  the  first  number  in  column  B,  of  the 
last  example,  from  each  number  in  column  A :  also 
from  each  number  in  column  C.  Proceed  in  like 
manner  with  each  of  the  remaining  numbers  of 
column  B. 


E.]        Tables  and  Questions  for  Practice.         25 


E.   Tables  and  Questions  for  Practice. 
TABLE  1. 


A 

B 

C 

D 

E 

F 

G 

H 

I 

1 

19 

18 

17 

16 

15 

14 

13 

12 

11 

2 

29 

28 

27 

26 

25 

24 

23 

22 

21 

3 

39 

38 

37 

36 

35 

34 

33 

32 

31 

4 

49 

48 

47 

46 

45 

44 

43 

42 

41 

5 

59 

58 

57 

56 

55 

54 

53 

52 

51 

6 

69 

68 

67 

66 

65 

64 

63 

62 

61 

7 

79 

78 

77 

76 

75 

74 

73 

72 

71 

8 

89 

88 

87 

86 

85 

84 

83 

82 

81 

9 

99 

98 

97 

96 

95 

94 

93 

92 

91 

1.  Add  2  to  each  number  in  column  A  of  Table 
1 ;  add  successively  3,  4,  5,  6,  7,  8,  9,  10,  to  the 
same  numbers. 

[In  this  case  the  pupil  may  be  asked  :  How  many  are  19  and 
2  ?  29  and  2  ?  etc. ;  19  and  3  ?  29  and  3  ?  etc.] 

Proceed  in  like  manner  with  each  of  the  remain- 
ing columns. 

2.  Add  2  to  each  number  in  line  1 ;  add  suc- 
cessively 3,  4,  5,  6,  7,  8,  9, 10,  to  the  same  numbers. 

Proceed  in  like  manner  with  each  of  the  remain- 
ing lines. 

3.  Subtract  2  from  each  number  in  column  A ; 
subtract  successively  3,  4,  5,  6,  7,  8,  9,  10,  from 
the  same  numbers. 

[In  this  case  the  pupil  may  be  asked  :   How  many  are  19  less 
2  ?  29  less  2  ?  etc. ;  19  less  3  ?  29  less  3  ?  etc.] 

Proceed  in  like  manner  with  each  of  the  remain- 
ing columns. 


26     Notation,  Addition,  and  Subtraction.     [§  1. 

4.  Subtract    2   from    each   number   in  line  1  ; 
subtract  successively  3,  4,  5,  6,  7,  8,  9,  10,  from 
the  same  numbers. 

Proceed  in  like  manner  with  each  of  the  remain- 
ing lines. 

5.  Find  the  sum  of  the  numbers  in  each  column. 

6.  Find  the  sum  of  the  numbers  in  each  line. 

TABLE  2. 


A 

B 

C 

D 

E 

1 

426 

6984 

976 

8432 

19798336 

2 

684 

7697 

2864 

6798 

11604186 

3 

799 

926 

1984 

1080 

3222720 

4 

231 

294 

962 

18006 

55494492 

5 

700 

7698 

111 

132 

167376 

6 

119 

8432 

555 

10832 

178316384 

7 

294 

9012 

7006 

69834 

681649674 

8 

371 

2984 

1080 

76497 

219087408 

9 

187 

6798 

9807 

984 

9275184 

10 

679 

832 

16 

16214 

80453868 

11 

138 

1981 

484 

1111 

8552478 

12 

291 

6462 

108 

5555 

46839760 

13 

545 

1268 

132 

7006 

63201126 

14 

379 

8006 

806 

9821 

24395364 

7.  Find  the  sum  of  the  numbers  in  each  column 
of  Table  2. 

8.  Find  the  sum  of  the  numbers  in  each  line. 

9.  Subtract  the  first  number  in  column  A  from 
the  first  number  in  column  B  ;  the  second  number 
in  column  A  from  the   second  number  in  column 
B,  and  so  on  to  the  ends  of  these  columns. 


E.]        Tables  and  Questions  for  Practice.         27 

1O.  Find  the  number  of  square  miles,  the  number 
of  inhabitants  in  1820,  and  in  1880,  in  each  of  the 
two  groups  of  states  given  below  ;  find  also  the 
number  of  United  States  soldiers  furnished  for  the 
civil  war  of  1861-65. 

Group  A.   (New  England  States.) 


States. 

Sq.  Miles. 

Population 
in  1820. 

Population 
in  1880. 

Soldiers 
furnished. 

Maine 

29,890 

298,335 

648,936 

72,114 

New  Hampshire 

9,005 

244,161 

346,991 

34,629 

Vermont 

9,135 

235,764 

332,286 

35,262 

Massachusetts 

8,040 

523,287 

1,783,085 

152,046 

Rhode  Island 

1,085 

83,059 

276,531 

23,699 

Connecticut 

4,845 

275,248 

622,700 

57,379 

Group  B.    (Middle  States  and  the  District  of 
Columbia.) 


States. 

Sq. 
Miles. 

Population 
in  1820. 

Population 
in  1880. 

Soldiers 
furnished. 

New  York 

47,620 

1,372,812 

5,082,871 

467,047 

New  Jersey 

7,455 

277,575 

1,131,116 

81,010 

Pennsylvania 

44,985 

1,049,398 

4,282,891 

366,107 

Delaware 

1,960 

72,749 

146,608 

13,670 

Maryland 

9,860 

407,350 

934,943 

50,316 

District  of  Co- 

lumbia 

60 

33,039 

177,624 

16,872 

11.  Find  the  number  of  square  miles,  and  the 
number  of  inhabitants  in  1880,  in  each  of  the  fol- 


28      Notation,  Addition,  and  Subtraction.    [§  1. 

lowing  groups  of  states  and  territories  ;  find  also 
the  number  of  United  States  soldiers  furnished  for 
the  civil  war  of  1861-65. 

Group  C. 


States. 

Sq.  Miles. 

Population 
in  1880. 

Soldiers 
furnished. 

Virginia 

40,125 

1,512,565 



West  Virginia 

24,645 

618,457 

32,068 

North  Carolina 

48,580 

1,399,750 

3,156 

South  Carolina 

30,170 

995,577 

- 

Georgia 

58,980 

1,542,180 

- 

Florida 

54,240 

269,493 

- 

Alabama 

51,540 

1,262,505 

2,576 

Mississippi 

46,340 

1,131,597 

545 

Louisiana 

48,420 

939,946 

5,224 

Texas 

262,290 

1,591,749 

1,965 

Group  D. 


States. 

Sq.  Miles. 

Population 
in  1880. 

Soldiers 
furnished. 

Michigan 

57,430 

1,636,937 

89,372 

Wisconsin 

54,450 

1,315,497 

96,424 

Ohio 

40,760 

3,198,062 

319,659 

Indiana 

35,910 

1,978,301 

192,147 

Illinois 

56,000 

3,077,871 

259,147 

Kentucky 

40,000 

1,648,690 

79,025 

Tennessee 

41,750 

1,542,359 

31,092 

Minnesota 

79,205 

780,773 

25,052 

Iowa 

55,475 

1,624,615 

76,309 

Missouri 

68,735 

2,168,380 

109,111 

Arkansas 

53,045 

802,525 

8,289 

E.]        Tables  and  Questions  for  Practice.         29 


Group 


States. 

Sq.  Miles. 

Population 
in   1880. 

Soldiers 
furnished. 

Nebraska 

76,185 

452,402 

3,157 

Kansas 

81,700 

996,096 

20,151 

Colorado 

104,500 

194,649 

4,903 

Nevada 

112,090 

62,265 

1,080 

California 

155,980 

864,694 

15,725 

Oregon 

94,560 

174,768 

1,810 

Group  F.    (Territories.) 


Territories. 

Sq.  Miles. 

Population 
in  1880. 

Soldiers 
furnished. 

Alaska 

577,399 

35,000 



Arizona 

113,020 

40,440 

- 

Dakota 

149,100 

135,177 

206 

Idaho 

84,800 

32,610 

— 

Montana 

146,080 

39,159 

- 

New  Mexico 

122,580 

119,565 

6,561 

Utah 

84,970 

143,963 

- 

Washington 

69,180 

75,116 

964 

Wyoming 

97,890 

20,789 

— 

12.  What  is  the  difference  between  the  number 
of   square   miles   in    Texas,  and  the  sum  of  the 
square  miles  in  New  York,  Pennsylvania,  Virginia, 
Wisconsin,  and  South  Carolina  ? 

13.  What  was  the  whole  number  of  inhabitants 
in   the  United   States   (including  the  District  of 
Columbia  and  the  Territories)  in  1880  ? 


30      Notation,  Addition,  and  Subtraction. 

14.  In  addition  to  the  number  of  United  States 
soldiers  sent  to  the  civil  war  of  1861-65,  from  the 
states  and  territories  given  above,  there  were  3,530 
Indian  troops  and  93,441  colored  troops.     How 
many  soldiers  were  furnished  in  all  ? 

15.  How  many  more  soldiers  were  sent  by  the 
Middle  States  than  by  the  New  England  States  ? 

16.  How  many  more  inhabitants  were  there  in 
the  New  England  States  in  1880  than  in  1820  ? 

17.  How  many  more  inhabitants  were  there  in 
the  Middle  States  and  the  District  of  Columbia  in 
1880  than  in  1820  ? 

18.  How  many  more  square  miles  are  there  in 
the  largest  state  than  in  the  smallest  ?  how  many 
more  inhabitants? 

19.  How  many  more  square  miles  are  there  in 
Colorado  than  in  Pennsylvania  ? 

20.  How  many  more  inhabitants  are  there  in 
Pennsylvania  than  in  Colorado  ? 

21.  What  is  the  difference  between  the  number 
of  square  miles  in  Texas  and  the  total  number  of 
square  miles  in  the  New  England  States  ? 


SECTION  II. 
MULTIPLICATION  AND   DIVISION. 

The  portions  of  this  section  that  are  preceded  by  a  star  (*) 
have  been  drawn  from  Section  VII.  of  the  Revised  Edition  of 
Colburn's  First  Lessons,  and  may  be  omitted  at  the  discretion  of 
the  teacher  by  pupils  who  have  studied  them  before. 

A.   Multiplication :    Examples  and  Problems, 
with  Remarks  and  Explanations. 

*1.  What  will  four  pounds  of  coffee  cost  at 
thirty-two  cents  a  pound  ? 

Four  pounds  of  coffee  will  cost  four  times  as 
much  as  one  pound,  or  four  times  thirty-two 
cents. 

32  =  3  tens +  2  units. 

4  times  32  =  4  times  3  tens +  4  times  2  units 
=  12  tens -I- 8  units 
=  120  +  8  =  128. 

Four  pounds  of  coffee,  then,  will  cost  128  cents  or 
11.28. 

*2.  Mr.  Wood,  the  grocer,  makes  a  profit  of  21 
cents  on  every  bag  of  meal  that  he  sells ;  what 
will  be  his  profit  on  4  bags  ? 
*3.  How  many  are  5  times  41  ? 
*4.  If   beefsteak  is  23  cents  per  pound,  what 
will  3  pounds  cost  ? 


32  Multiplication  and  Division.          [§  2. 

*6.  How  many  are  4  times  52  ? 

52  =  5  tens  +  2  units. 

4  times  52  =  4  times  5  tens  +  4  times  2  units 
=  20  tens +  8  units 
=  200  +  8  =  208. 
To  save  space  we  may  write  our  work  thus  : 

Or,  still  more  briefly  :     f      52 
We   may  briefly  de-J        4 
scribe  our  work  as  fol-      208 
lows  :  4  times  2  are  8  ; 


52 


8  =  4x2  units. 
200  =  4x5  tens. 


OQQ  _    x  co  we  write  down  the  8  :  4  times 

5  are  20  ;  we  write  down  the 
20  to  the  left  of  the  8  and  get  208  for  an  answer. 
*6.  Multiply     42          61          53          82 
by          JB          _4          _3          _5 

126 

*7.  When  we  have  to  perform  an  example  in 
multiplication  it  is  sometimes  convenient  to  call 
the  number  which  we  are  asked  to  multiply  the 
MULTIPLICAND,  the  number  by  which  we  are  to 
multiply  the  MULTIPLIER,  and  the  result  obtained 
by  our  work  the  PRODUCT. 

Thus,  in  example  5  the  multiplicand  is  52,  the 
multiplier  is  4,  and  the  product  is  208. 

Name  the  multiplicand,  the  multiplier,  and  the 
product  in  the  examples  of  6. 

*8.  If  one  barrel  of  sugar  costs  19  dollars,  how 
much  will  5  barrels  cost? 

The  answer  will  be  5  times  19  dollars,  and  we 
are  to  find  out  how  many  dollars  this  will  make. 
19  =  1  ten +9  units. 


A.]  Multiplication.  33 

5  times  19  =  5  times  1  ten +  5  times  9  units 

=  5  tens  4-  45  units 
=  5  tens +  4  tens +  5  units 
=  9  tens +  5  units 
-95. 

Five  barrels  of  sugar,  then,  at  19  dollars  a  bar- 
rel will  cost  95  dollars. 

*9.  If  a  railway  train  goes  36  miles  in  an  hour, 
how  far  will  it  go  in  3  hours  ? 
*1O.  How  many  are  5  times  64  ? 
*11.  Mr.  Eeardon  gets  25  cents  an  hour  for  his 
work  ;  how  much  does  he  get  for  8  hours'  work  ? 
how  much  for  4  hours'  work  ? 
*12.  How  many  are  6  times  78  ? 
78  =  7  tens +  8  units. 

6  times  78  =  6  times  7  tens +  6  times  8  units 

=  42  tens +  48  units 
=  42  tens +  4  tens +  8  units 
=  46  tens  +  8  units 
=  460  +  8 
=  468.  Answer. 
To  save  space  we  may  78 

write  our  work  thus  :  6 

48  =  6x8  units. 
420  =  6x7  tens. 


468-6x78. 
*13.  a.  Multiply  29 
by  '_7 

63  =  7x9  units. 
140  =  7x2  tens. 

Answer:     203  =  7x29. 


34  Multiplication  and  Division.         [§  2. 

6.  How  many  are  36  x  5  ?  43  x  8  ?  57  x  6  ? 

*14.  Multiply     84  ^       Instead  of  writing  down 

by  6     the  24  and  the  480  sepa- 

24  v  rately,  we  may  add  them 

480     together  in  our  heads  as 

TT7      we  go  on  and  say:  6  times 
Answer:        504 )  A 

4  are  24,  or  2  tens  and  4 

units  ;  we  set  down  the  4  and  save  the  2  tens  to 
add  in  with  other  tens.  6  times  8  tens  are  84 
48  tens ;  these  with  the  2  tens  saved  over  6 
make  50  tens,  and  we  write  down  the  50  to  504 
the  left  of  the  4. 

*15.  Multiply     96^1       4  times  6  are  24;  set 
by  4  I  down  the  4  and  save  the 

2.  4  times  9  are  36  ;  add 
in  the  2  that  were  saved 

and  set  down  the  result,  38,  to  the  left  of  the  4. 
*16.  Multiply     85^       7  times  5  are  35;  set 
by  7  y  down  the  5  and  save  the 

3.  7  times  8  are  56  ;  add 
in  the  3  and  set  down  the 

result,  59,  to  the  left  of  the  5. 

*17.  Multiply        49        28        37        65 
by  _T_       _8       _5       _j> 

343      "224      185      585. 

Name  the  multiplicand,  the  multiplier,  and  the 
product  in  each  case. 

*18.  What  will  six  pounds  of  chocolate  cost  at 
38  cents  per  pound  ? 

*19.  At  19  cents  a  pound,  what  will  a  roasting 
piece  of  beef  weighing  8  pounds  cost  ? 


595  I  3. 

'      ir 


A.]  Multiplication.  35 

*2O.  If  7  men  can  dig  a  ditch  in  24  days,  how 
long  would  it  take  1  man  to  do  it  ? 

*21.  If  one  barrel  of  vinegar  holds  32  gallons, 
how  much  vinegar  can  be  put  into  6  barrels  ?  how 
much  into  8  barrels  ? 

*22.  If  one  barrel  of  flour  contains  196  pounds, 
how  many  pounds  will  there  be  in  four  barrels  ? 

196  =  1  hundred  +  9  tens  +  6  units. 
4x196  =  4x1  hundred  +  4x9  tens  +  4x6  units 
=  4  hundred  +  36  tens  +  24  units 
=  400  +  360  +  24 
-784. 

Four  barrels  of  flour,  then,  will  contain  784 
pounds. 

We  may  save  space  by  arranging  our  work 
thus  : 

196 


24  =  4x6  units. 
360  =  4x9  tens. 
400  =  4x1  hundred. 

~784  =  4x196. 

*23.  How  many  are  3  times  218  ? 
218 
_3 

24  =  3x8  units. 
30  =  3x1  ten. 
600  =  3x2  hundred. 
Answer:  "654  =  3x218. 
Multiply  341         208         121         610 

by  '  6  3  8  6 


36  Multiplication  and  Division.  [§  2. 

*24.  Mr.  Blake  paid  three  dollars  and  twenty- 
five  cents  apiece  for  six  chairs  ;  how  much  did  the 
whole  cost  him  ? 

*25.  If  one  barrel  of  flour  is  worth  seven  dollars 

and  forty  cents,  how  much  are  six  barrels  worth  ? 

*26.  Multiply  243 ^       Instead  of  writing  down 

by  8     the  24,  the  320,  and  the 

24  I  1600,  separately,  we  may 

320  f  ^^  them  together  as  we 

1600     g°  on  with  our  work,  and 

1044]  sav:   ^   times  3  are   24; 

we   set   down   the   4  and 

save  the  2  tens.  8  times  4  tens  are  32  tens,  and 
these  with  the  2  tens  saved  over  make  34  tens,  or 
3  hundred  and  4  tens ;  we  set  down  the  4  tens  and 
save  the  3  hundred.  8  times  2  hundred  are  243 
16  hundred,  and  these  with  the  3  hundred  8 

make  19  hundred,  which  we  set  down  to  the     1944 
left  of  the  figures  that  we  already  have. 
In -like  manner 

Multiply          416         123        321        643 
by  7452 

*27.  Multiply~134by7T~ 

After  you  have  learned  how  to  do  such  examples 
as  those  of  No.  26  correctly  and  quickly,  you  may 
tell  as  briefly  as  possible  what  you  actually  do  in 
multiplying  two  numbers  together,  thus  : 
134^1       7  times  4  are  28  ;  we  set  down  the  8  and 

7  I  carry  the  2.     7  times  3   are  21,  and  2  are 

938  [  23  ;  we  set  down  the  3  and  carry  the  2.     7 

J  times  1  are  7,  and  2  are  9 ;  we  set  down  the  9. 


A.]  Multiplication.  37 

*28.  Multiply     2196     1419     6021     8206 
by         5  368 

Answer:         10980 
*29.  10  times  36  are  how  many  ? 
We  know  that  10  times  36  are  the  same  as  36 
times  10.     Therefore  10  times  36  =  36  tens  or  360. 
10  times  49  are  how  many  ?     Answer  :  490. 
10  times  64  are  how  many  ?     Answer  :  640. 
10  times  90  are  how  many  ?     Answer  :  900. 
10  times  16  are  how  many  ?     Answer :  160. 
10  times  83  are  how  many  ?     Answer  :  830. 
You  have  just  multiplied  different  numbers  by 
10  ;  writing  down  each  of  these  numbers,  and  un- 
derneath it  10  times  the  number  we  have  : 

49          64          90          16          83 
490        640        900        160        830 
What  do  you  do,  then,  when   you   multiply  a 
number  by  10  ? 

*30.  20  times  37  are  how  many  ? 
20x37  =  10x2x37 

=  10x74  =  740.  Answer. 
*31.  Multiply         68        49        36        51 
by  '  30        40        50        70 

Answers:  2040     1960     1800     3570. 

-Notice  that  the  first  answer  may  be  obtained 
by  placing  a  zero   after  3  times  68,  the  second 
by  placing  a  zero  after  4  times  49,  the  third  by 
placing  a  zero  after  5  times  36,  and  so  on. 
*32.  a.  Multiply  46          46          46          46 

by          100        200        300        700 

Answers:      4600      9200    13800    32200. 


88  Multiplication  and  Division.  [§  2. 

Notice  that  the  first  answer  may  be  obtained  by 
writing  two  zeros  after  1  x  46,  the  second  by  writ- 
ing two  zeros  after  2  x  46,  etc.,  etc. 

6.  Multiply          64         82  76          98 

by  600        500          800        400 

Answers:        38400    41000      60800    39200. 
*33.  How  many  are  24  times  36  ? 
36 
24  =  20+  4. 


144  =  36  x   4. 
720  =  36x20. 


864  =  36x24. 

*34.  How  many  are  18  times  23  ? 

23 

_18  =  10+  8. 
184  =  23  x  8. 
230  =  23x10. 


414  =  23x18. 

*35.  If  butter  is  worth  28  cents  a  pound,  what 
is  the  value  of  a  tub  containing  35  pounds  ? 

Answer :  980  cents,  or  $9.80. 

*36.  What  is  the  cost  of  24  hoes  at  58  cents 
each  ? 

*37.  Mr.  Jones  bought  32  hens  at  96  cents 
apiece  ;  how  much  did  he  pay  for  them  all  ?  He 
then  sold  21  of  them  at  $1.10  apiece  ;  how  much 
did  he  get  for  these  ? 

*38.  If  a  man,  who  is  draining  a  field,  can  dig 
48  feet  of  ditch  in  one  day,  how  much  can  he  dig 
in  22  days  ? 


A.]  Multiplication.  39 

*39.  I  bought  42  boxes  of  soap  each  containing 
18  bars  ;  how  many  bars  of  soap  were  there  in  all? 
*40.  Multiply        98         69         73        46 
by  47        J59         13        54 

*41.  After  you  have  learned  to  multiply  cor- 
rectly, you  may  shorten  your  written  work  as  much 
as  possible. 

Multiply  49  by  27. 

49  49 

27         or,  more  briefly,  27 

343-49x    7.  343 

_980  =  49x20.  98 

1323  =  49x27.  1323. 

You  may  omit  the  zero  after  the  98,  if  you  are 
careful  to  show  that  the  98  represents  98  tens,  by 
moving  the  number  one  step  to  the  left,  so  that  the 
8  may  stand  in  the  column  of  tens  and  not  in  the 
column  of  units. 

2  times  6  are  12 ;  we 
set  down  the  2  and  carry 
the  1.  2  times  7  are  14, 
and  the  1  make  15.  5 
times  6  are  30 ;  we  set 
down  the  zero  in  the  col- 


*42.  Multiply     76 
by  52 


380 
3952 


umn  of  tens  and  carry  the  3.     5  times  7  are  35, 
and  the  3  make  38,  etc.,  etc. 

*43.  Multiply    71       48       64      49       83 
by          19      12       T2      _38       25 

Name  the  multiplicand,  the  multiplier,  and  the 
product,  in  each  case. 


196 

48 

or, 
more 
briefly, 

1568  =  196  x    8. 
7840-196x40. 

40  Multiplication  and  Division.  [§  2. 

*44.  If  a  barrel  of  flour  contains  196  pounds, 
how  many  pounds  are  there  in  48  barrels  ? 

196 

48 

1568 

784 

9408-196x48.  9408. 

*45.  If  coal  is  worth  four  dollars  and  sixty-four 
cents  a  ton  by  the  cargo,  what  will  a  cargo  of  three 
hundred  and  twenty-seven  tons  cost  ? 

If  one  ton  costs  464  cents,  327  tons  will  cost 
327  times  464  cents.  We  are,  then,  to  multiply 
464  by  327. 

327-300  +  20  +  7. 

464  464 

327  327 

3248  =  464  x     7.      or,  3248 

9280  =  464  x    20.    more  928 

139200  =  464x300.  briefly,     1392 
151728  =  464x327.  151728 

The  whole  cargo  will  cost,  then,  151728  cents, 
or  1517  dollars  and  28  cents. 


*46.  a.  Multiply  318 
by          425 

Multiply 

by 

Answer  : 
J23        417 
196        391 

406 
719 

Answer  : 
b.  Multiply 

by 

1590 
636 

1272 

3654 
406 

2842 

135150. 

873        i 
614        ] 

291914. 
803 
516 

A.]  Multiplication.  41 

*47.  a.  Multiply  456  by  204. 

204  =  2  hundred  +  0  tens  +  4  units. 

-200  +  4. 

456                                or,  456 

204                               more  204 

1824- 456  x     4.        briefly,  1824 

91200-456x200.  912          . 

93024-456x204.  93024. 

b.  Multiply     471        822        583  649 

by           306         507         109  408 

Answer:  144126 

*48.  In  multiplying  one  number  by  another, 
you  have  learned  to  write  down  the  number  to  be 
multiplied  (the  "  multiplicand ")  over  the  multi- 
plier ;  and,  beginning  at  the  right,  to  multiply  the 
multiplicand  by  each  figure  of  the  multiplier  sepa- 
rately ;  and  to  write  down  your  results  one  over 
another,  so  that  the  right-hand  figure  of  each  shall 
be  directly  under  that  figure  of  the  multiplier 
which  was  used  to  obtain  this  result.  By  adding 
together  the  results  thus  arranged  you  get  the 
product  required.  Thus : 

Multiply  8164^  In  this  case  8164  is  the 
multiplicand  and  4821  is 


•  by      4821 

8164 
16328 
65312 
32656 


39358644 


the   multiplier.      The  re- 
.  suit  obtained  by  multiply- 
ing 8164  by  the  1  is  8164, 
and  is  set  down  so  that  its 
right-hand  figure  4  is  di- 
rectly  under   the    1.      In 
the  same  way  the  result,  16328,  obtained  by  multi- 


42  Multiplication  and  Division.  [§  2. 

plying  8164  by  the  2,  is  set  down  so  that  its  right- 
hand  figure  is  directly  under  the  2,  etc.,  etc. 
Multiply       3142       7162      6824      5691 
by  693       8237       9036       2007 

49.  Show  that  93  multiplied  by  587  is  the  same 
as  587  multiplied  by  93. 

In  this  example  and  in  others  like  it  the  product  is  the  same, 
whether  the  first  number  is  chosen  as  the  multiplicand  and  the 
second  as  the  multiplier,  or  the  second  is  chosen  as  the  multipli- 
cand and  the  first  as  the  multiplier. 

50.  Find  the  product  of  1728  and  132,  first  by 
using  1728  as  the  multiplicand  and   132  as  the 
multiplier,  and  second  by  using  132  as  the  multi- 
plicand and  1728  as  the  multiplier. 

Which  number,  the  larger  or  the  smaller,  should 
you  select  for  a  multiplicand  in  order  to  get  the 
product  with  the  least  labor  ? 

51.  Find  the  product  in  each  of  the  following 
cases,  selecting  for  your  multiplier  in  each  case  the 
number  which  enables  you  to  get  your  result  with 
the  least  labor :  - 

29x384  1142x38  17x684 

497x36  79x11238  45x125 

52.  Find  the  product  of  360  and  27. 

360  =  10  x  36,  therefore  360  x  27  =  10  x  36  x^27. 
The  work  may  be  arranged  as  follows :  — 
Either  Or 

27  360 

360  27 

162T  252 

81  72 

9720  9720 


A.]  Multiplication.  43 

53.  Find  the  product  of  (»  13600  and  64  and 
(6)  of  96000  and  127. 

(a)  13600  (6)  127 

64  96000 

544~  ~762~~ 

816  1143 


870400.  12192000. 

54.  Find  the  product  of 

(a)  386000  and  86.          (c)  124000  and  42. 
(6)  96  and  121000.         (d)  7900  and  132. 

55.  The  railroad  from  Providence  to  Worcester 
is  44  miles  long;  there  are  1760  yards  in  a  mile; 
how  many  yards  of  track  are  there  between  Provi- 
dence and  Worcester? 

56.  There  are  480  sheets  in  a  ream  of  paper ; 
how  many  sheets  are  there  in  605  reams  ? 

57.  A  man  owing  $105,760  gives  in  payment 
180  acres  of  land  worth  $65  an  acre  and  $5000  in 
money.     How  much  remains  unpaid  ? 

58.  How  much  more  do  112  cords  of  wood  at 
$5.25  a  cord  cost  than  75  yards  of  cloth  at  $5  a 
yard? 

59.  How  many  miles  has  a  railroad  conductor 
traveled  in  3  years  who  has  gone  over  the  road 
from  Louisville  to  Nashville  —  185  miles  —  once 
each  day  ? 

60.  A  flour  dealer  bought  42  barrels  of  flour 
for  $210.     He  sold  £  of  it  at  $4.75  a  barrel,  and 
the  remainder  at  $6  a  barrel.     How  much  did  he 
gain  ? 


44  Multiplication  and  Division.          [§  2. 

61.  How  much  will  17  bales  of  cloth  cost  at  22 
cents  a  yard  if  each  bale  contains  50  pieces  and 
each  piece  contains  38  yards  ? 

62.  I  bought  350  acres  of  land  at  $17.60  an 
acre  and  sold  the  whole  at  $16  an  acre.     Did  I 
gain,  or  lose,  and  how  much  ? 

63.  A  man  bought  3  tons  of  hay  at  $18.50  a 
ton,  12  barrels  of  apples  at  $3.50  a  barrel,  and  a 
suit  of  clothes  worth  $50.     In  payment  he  gave 
17  barrels  of  flour  at  $4.75  a  barrel  and  the  bal- 
ance in  money.     How  much  money  did  he  give  ? 

64.  How  much  less  is  608  times  3607  than  6075 
times  10476? 

65.  Find  the  cost  of  each  of  the  following  items 
and  add  the  results  together :  — 

42  bushels  of  turnips  at  $0.37  per  bushel. 

213  pounds  of  butter  at  $0.23  per  pound. 

126  pounds  of  coffee  at  $0.33  per  pound. 

24  boxes  soap  at  $4.25  per  box. 

12  pitchers  at  42  cents  apiece. 

16  doz.  lamp  chimneys  at  80  cents  per  doz. 

450  Ibs.  of  rice  at  9  cents  per  Ib. 

66.  Find  the  cost  of    the  articles  which  Mr. 
Slade  bought  for  his  store  when  he  was  last  in 
Boston. 

He  bought : 

60  yards  of  nun's  veiling  at  75  cents  per  yard. 

4  doz.  pair  hose  at  50  cents  a  pair. 

72  yds.  sateen  at  37  cents  a  yard. 

18  doz.  towels  at  $3  per  doz. 

200  spools  sewing  silk  at  8  cents  per  spooL 


A.] 


Multiplication. 


45 


67.  The  number  of  full  pages  in  Hawthorne's 
Wonder- Book   is    178  ;    the    average   number   of 
words  on  a  page  is  307.     How  many  words  are 
there  in  all  ? 

68.  During  the  year  ending  June  30,  1887,  the 
United    States    Post    Office    Department    bought 
25,500  leather  mail  bags  at  an  average  cost  of 
$264  a  hundred  ;  what  was  the  entire  cost  ? 

69.  In  1880  the  population  of  Cleveland,  Ohio, 
was  160,142,  and  the  debt  of  the  city  (amount  of 
money  owed  by  it)  was  such  that  all  of  it  could 
have  been  paid  if  each  inhabitant  had  contributed 
125.45  ;  what  was  the  debt? 

70.  Below  is  given  the  population  and  debt  per 
person  *  in  1880  of  each  of  the  nine  largest  cities 
of  the  United  States  ;  what  was  the  entire  debt  of 
each  city  ? 


Population. 

Debt  per 
person. 

1 

New  York,  N.  Y. 

1,206,299 

$90.69 

2 

Philadelphia.  Pa. 

847,170 

19.18 

3 

Brooklyn,  N.  Y. 

566,663 

67.13 

4 

Chicago,  111. 

503,185 

25.42 

5 

Boston,  Mass. 

369,832 

77.90 

6 

St.  Louis,  Mo. 

350,518 

65.18 

7 

Baltimore,  Md. 

332,313 

81.55 

8 

Cincinnati,  Ohio. 

255,139 

86.00 

9 

San  Francisco,  Cal. 

233,959 

13.12 

*  By  debt  per  person  is  meant  the  amount  that  each  inhabitant 
would  have  owed  if  the  entire  debt  had  been  divided  equally 
among  all  the  inhabitants. 


46 


Multiplication  and  Division.         [§  2. 


B.  Bills. 

NOTE.  — On  Dec.  11,  1887,  John  H.  Brown 
bought  of  Jones,  Smith  &  Co.,  of  Chicago,  111., 
24  copies  of  the  Household  Edition  of  Long- 
fellow's Poems  at  $1.17  a  copy,  and  36  copies  of 
Colburn's  Arithmetic  at  29|  cents  a  copy.  He 
received  with  the  books  the  following  BILL  : 


Mr.  John  H.  Brown 


CHICAGO,  ILL.,  Dec.  11,  1887. 


£0  JONES,  SMITH  &,  CO.,  Dr. 


24  Longfellow's  Poems,  Hid  Ed.,  @  $1.17 
86  Colburn's  Arithmetic,                 "    29\c. 

$28 
10 

08 
71 

$88 

79 

On  Dec.  24  Mr.  Brown  took  the  bill,  with  the 
money  that  he  owed,  to  Jones,  Smith  &  Co.'s 
cashier,  H.  M.  Twitchell,  and  received  the  bill 
back  again,  RECEIPTED,  as  shown  below. 

CHICAGO,  ILL.,  Dec.  11,  1887. 

Mr.  John  H.  Brown 

&0  JONES,  SMITH  &,  CO.,  Dr. 


$4  Longfellow's  Poems,  If'ld  Ed.,  @  $1.17 
86  Oolburn's  Arithmetic,  "    %9\c. 


08 

7JL 

79 

Received  payment,        JONES,  SMITH  &  Co. 
Dec.  24,  1887.  per  H.  M.  TWITCHELL. 


$28 
10 


B.]  Bills.  47 

1.  Copy  and  complete  the  following  bills  : 
A. 

BOSTON,  MASS.,  DEC.  22,  1887. 


Mr.  A.  B.  Cole 


DEE  &,  ELA,  Dr. 


10%  Rolls  of  Paper,                       @ 

50c. 

3       "      "  Frieze,                       " 

75c. 

56    Feet  of  Moulding,                   « 

8c. 

Hanging  11  Rolls  of  Paper,          " 

37\c. 

"           3     "       «  frieze,         " 

37\c. 

"        56  Ft.  of  Moulding,       « 

3c. 

Taking  off  Paper 

. 

* 

00 

Car  Fares    ..... 

.         . 

48 

Received  Payment, 

DEE  &  ELA, 

B. 

CAMBRIDGE,  MASS.,  Jan.  2,  1888. 
Mr.  A.  B.  Jones 

&0  THE  CAMBRIDGE  GAS-LIGHT  COMPANY,  Dr. 


For  Gas  consumed  during  the  quarter 
ending  this  day  .... 

21500  cubic  feet  at  20  cents  per  hundred, 
Discount  if  paid  before  Jan.  26, 
1888,  at  25  cents  per  thousand 


Received  payment  for  the  Company, 

ADOLPH  VOOL. 
Date,  Jan.  12,  1888. 


48 


Multiplication  and  Division.          [§  2. 


Mr.  C.  L.  Wood 


a 

BOSTON,  MASS.,  Sept.  19,  1884. 
Of  JOEL  GOLDTHWAIT  &,  CO. 


26  yds.  Art  Kidderminster,        @ 

Sewing  and  Fitting,  "  We. 

26  yds.  Lining,  "          12\c. 


Paid  Oct.  1,  1884. 

JOEL  GOLDTHWAIT  &  Co. 

per  A.  M.  LOOMIS. 


D. 

CAMBBIDGEPORT,  MASS.,  Oct.  16,  1886. 

Mr.  O.  F.  Ambrose 

fto  GEORGE  F.  RICKER  &  CO.,  Dr. 


Sept. 


Oct. 


8    Taking  up  25  yds.  of  Carpet, 

©      2c. 
8    Cleaning  25  yds.  of  Carpet, 

@      4*. 

8         "        1  Rug 
8  Laying  28  yds.  of  Carpet,  @  J^c. 
8        "       25    "     "          "      "   4c. 
18  Renovating  8  pairs  of  Pillows, 

@  $1.00. 
18    Washing  8  Pillow  Ticks,  @  25c. 

15    Credit  by  check  received 
Balance  due, 


B.] 

Bills.                                49 

E. 

BOSTON,  Oct.  2,  1887. 

N.  H.  Cooler,  Esq. 

Bought  of  L.  T.  PRICE  &/  CO., 

IMPORTERS  AND  GROCERS. 

25  Ibs.  Coffee, 

@     35c. 

8   "     Tea, 

"      75c. 

1  doz.  bottles  Salad  Oil, 

@  75c.  per  bottle. 

3    "     boxes  of  Soapine,  @  lie.  per  box. 

9    "     Sapolio, 

@  96c.  per  doz. 

lt\  Ibs.  Royal  Baking  Powder,        @  lf.2c. 

18  boxes  Gelatine, 

44    13c. 

24  Ibs.  Macaroni, 

"    13c. 

16    "  Spaghetti, 

"    We. 

8  doz.  cans  Tomatoes, 

@  $1.60  per  doz. 

17  Ibs.  Rice, 

@     8c. 

17  boxes  Silver  White, 

u     8c. 

16  Ibs.  Am.  Oatmeal, 

"  4\c. 

24    "  Rye  Meal, 

"  $&• 

29    "  Granulated  Sugar 

"  7ic. 

16    "  Pearl  Tapioca, 

"   Plc- 

12  gals.  Dark  Molasses, 

"  ^c. 

15  Ibs.  Am.  Chocolate, 

"    We. 

13    "    Cream  oj  Tartar, 

"  62c. 

\  Ib.  B.  Pepper, 

«  We. 

3  qts.  S.  S.  P.  Olives, 

u    RQc 

j  Ib.  Ground  Mustard, 

"  40c" 

Received  payment, 


L.  T.  PRICE  &  Co. 

fy  S.  L.  T. 


50  Multiplication  and  Division.          [§  2. 

F. 

BOSTON,  MASS.,  Oct.  4,  1887. 
Messrs.  Selwood  $  Sons 

Ett  ftcct.  but!)  BABK,  STEELE  &  CO.,  Dr. 


188 

June 

4 

140  tons  Steel  Rails, 

@  $82.00 

u 

15 

28  doz.  Axes, 

9.25 

July 

5 

175  tons  Zinc  Dross, 

"     42-00 

Aug. 

20 

487  Ibs.  Zinc, 

"       *05\ 

Sept. 

80 

814  cwt.  Lead, 

6.85 

Cr. 

July 

2 

500  bbh.  Flour, 

"      4M 

" 

17 

825  bu.  Wheat, 

.95 

Aug. 

24 

Draft  on  New  York 

. 

875  00 

Sept. 

8 

117  SharesofR.R. 

Stock, 

@  $85.00 

Balance  due,  . 

• 

Received  payment, 

BARR,  STEELE  &  Co. 

by  JOHN  SMALL. 

2.  Make  out  a  bill  dated  to-day  against  the  City 
of  Boston  for  15  J  hours  of  work  done  by  yourself 
at  24  cents  an  hour. 

3.  Make   out  a  bill  for  each  of  the  following 
cases :  — 

a.  Cambridge,   Mass.,   Dec.    15,   1887.     O.  B. 
French  made  for  Rufus  Bullock  1056  ft.  of  board 
walk  at  17i  cents  a  foot.     Paid  Dec.  20,  1887. 

b.  Boston,   Mass.,   Apr.  27,  1885.     Houghton, 


C.]  Division.  51 

Mifflin  &  Co.  bound  for  B.  O.  Peirce  25  volumes 
of  the  Waverley  novels  at  $1.25  a  volume.  Paid 
Apr.  27,  1885. 

c.  Chicago,  111.,  June  1,  1888.     James   Brown 
owes  John  Smith,  stable-keeper,  for  board  of  horse 
from  May  1  to  June  1,  1888,  $26.52  ;  for  hack, 
May  2,  $1.50  ;  for  hack,  May  4,  $3.00  ;  for  use  of 
carryall,  May  10,  $1.50;  for  1  halter,  May  15, 
$1.25.     Paid  May  15,  $5.00,  May  25,  $10.00,  and 
the  balance  on  June  10. 

d.  Indianapolis,  July  1,    1884.     On   June  24, 
1884,  A.  S.  Jones  bought  of  Richardson  &  Bacon 
8  tons  of  Excelsior  coal  at  $6.00  a  ton,  and  12  tons 
of  Draper  Furnace  coal  at  $5.75  a  ton  ;  the  charge 
for  putting  into  the  cellar  was  25  cents  a  ton. 

C.   Division:   Examples  and  Problems,  with 
Remarks  and  Explanations. 

*1.  If  one  barrel  of  apples  costs  3  dollars,  how 
many  barrels  can  you  buy  for  96  dollars  ? 
We  are  to  divide  96  by  3. 

96  =  9  tens  +  6. 


=  3  tens  +  2 
-32. 

You  can,  then,  buy  32  barrels  of  apples,  at  3 
dollars  a  barrel,  for  96  dollars. 

Divide  84  by  4  ;  48  by  2  ;  and  63  by  3. 
*2.   A  man,  who  wished  to  pay  a  bill  of  2  dollars 
and  48  cents  in  a  neighboring  town,  bought  the 


52  Multiplication  and  Division.  [§  2. 

money's  worth  of  2-cent  stamps,  and  sent  the 
stamps  in  a  letter  to  his  creditor;  how  many 
stamps  did  he  buy?  We  are  to  divide  248  by  2. 

248  =  2  hundreds +  4  tens +  8. 
248-^2  =  2  hundreds -r 2 4-4  tens-f2  +  8-r2 
=  1  hundred  +  2  tens  +  4 
=  124.  Answer. 

Divide  369  by  3  ;  484  by  4 ;  and  624  by  2. 
*3.  If  you  divide  92  counters  among  4  boys,  how 
many  will  each  receive  ? 


92  counters  =9  rows +  2  counters. 


There  are  evidently  rows  enough  to  give  each 
boy  2  full  rows  ;  now  dividing  the  remaining  1  row 
and  2  counters,  or  12  counters,  among  the  4  boys, 
each  will  receive  3  counters. 

Therefore,  each  boy  will  receive  in  all  2  rows  +  3 
counters,  or  23  counters. 

*4.  If  you  divide  87  counters  among  3  girls, 
how  many  will  each  receive  ? 


•*••••••• 

•  ••••••••  I 

•  ••••••••  )>  87  counters  =8  rows  +7  counters. 

•  ••••••••  I 


C.]  Division.  53 

There  are  evidently  rows  enough  to  give  each 
girl  2  full  rows;  now  dividing  the  remaining  2 
rows  and  7  counters,  or  27  counters,  among  the  3 
girls,  each  will  receive  9  counters. 

Therefore,  each  girl  will  receive  in  all  2  rows  4-  9 
counters,  or  29  counters. 

*5.  Gertrude,  Herbert,  and  Edith  went  out 
chestnutting  together,  agreeing  to  divide  equally  all 
that  they  should  find:  when  they  reached  home 
they  counted  out  their  chestnuts  into  heaps  of  ten 
each,  and  found  that  they  had  8  heaps  and  4  chest- 
nuts over,  that  is,  84  in  all;  how  many  chestnuts 
should  each  receive? 

If  you  solve  this  problem  by  the  aid  of  counters, 
using  a  counter  to  represent  a  chestnut,  you  will 
find  that  each  should  receive  28  chestnuts. 

*6.  Mr.  Smith,  Mr.  Jones,  and  Mr.  Robinson 
cultivated  together  a  vegetable  garden :  beside 
supplying  their  families  with  vegetables,  they  sold 
57  dollars'  worth  and  divided  the  money  ;  how 
much  did  each  receive? 

*7.  When  we  have  to  perform  an  example  in 
division,  it  is  sometimes  convenient  to  call  the 
number  which  we  are  asked  to  divide  the  DIVIDEND, 
the  number  by  which  we  are  to  divide  the  DIVISOR, 
and  the  result  obtained  by  our  work  the  QUOTIENT. 

Thus,  in  Example  3  the  dividend  is  92,  the 
divisor  is  4,  and  the  quotient  is  23. 

Name  the  dividend,  the.  divisor,  and  the  quotient 
in  the  Examples  of  1  and  2 ;  multiply  the  divisor 
by  the  quotient  in  each  case  and  see  if  your  result 
is  the  same  as  the  dividend. 


54 


Multiplication  and  Division.          [§  2. 


*8.  Divide  96  by  4. 

96  =  9  tens +  6  units. 

4  goes  into  9  tens  2  tens  times  and  leaves  an 
extra  ten,  which  added  to  the  6  units  makes  16 
units.  4  goes  into  16  units  4  times;  therefore  4 
goes  into  96 

2  tens  times  +  4  times  or  24  times. 

We  can  save  space  by  arranging  our  work  thus : 


4)96(20  +  4. 
80  =  4x20. 

16  remainder. 
16  =  4x4. 

0  remainder. 


We  may  say  4  goes  into 
9  tens  2  tens  times;  we 
set  down  the  2  tens  to  the 
right  of  the  dividend  and 
subtract  2  tens  times  4  or 
80  from  the  dividend. 


The  remainder  is  16.  4  goes  into  16  4  times ;  we 
set  down  the  4  to  the  right  of  the  dividend  and 
subtract  4  times  4  from  our  16.  The  remainder 
is  nothing.  Our  quotient,  then,  is  20  +  4  or  24. 

In  this  example  which  is  the  dividend  ?  which 
the  divisor  ?  which  the  quotient  ? 


*9.  Divide  78  by  3. 
3)78(20  +  6.  " 

60  =  3x20. 

18  =  remainder. 
18  =  3x6. 


Divide  52  by  4. 

4)52(10  +  3. 

40  " 

12 
12 


0  =  remainder. 

Answer :  26.  Answer :  13. 

Divide  36  by  2 ;  54  by  3 ;  72  by  4 ;  and  95 

by  5.     Name  the  dividend,  the  divisor,  and  the 

quotient,  in  each  case  ;  multiply  the  divisor  by  the 

quotient  and  compare  the  result  with  the  dividend. 


C.]  Division.  55 


*10.  Divide  532  by  4. 


4)532(100  +  30  +  3. 
400-4x100. 


4  goes  into  5  hundred  1 
hundred    times ;     we    set 


down  100  in  the  quotient 
=  remainder.  ,        .  -i    i       i     i 

190  — ji  OA  land    subtract    1    hundred 

f  times  4  or  400  from  532 
12  =  remainder. 

12  =  4x3. 


and    get    132    for    a  re- 
mainder.    4  into  13  tens 
0  )  goes  3  tens  times ;  we  set 

down  30  in  the  quotient  and  subtract  30  times  4  or 
120  from  132  and  get  12  for  a  remainder.  4  into 
12  goes  3  times  ;  we  set  down  the  3  in  the  quotient 
and  subtract  3  times  4  or  12  from  12  and  get 
nothing  for  a  remainder.  Our  quotient,  then,  is 

100  +  30  +  3,  or  133. 

Divide  423  by  3  ;  738  by  6  ;  314  by  2 ;  and 
854  by  7. 
*11.  Divide  156  by  4. 


4)156(30  +  9. 
120=7x80. 


4  is  not  contained  in  1 ; 
we    therefore    put   the    1, 


which    stands   for  1   hun- 
36  =  remainder.       \  ,     ,      .  ,    -  .  . 

Q£  —  A     Q  I   c*red>  with  5  tens  making 

OO  —  4  X  y.  l-ir.  4*1         -ir, 

15   tens.     4  into  15  tens 
J  goes   3  tens  or  30  times  ; 

we  write  the  30  in  the  quotient  and  subtracting  30 
times  4  or  120  from  156  we  get  36  for  a  remainder. 
4  into  36  goes  9  times ;  we  write  the  9  in  the 
quotient  and  subtracting  9  times  4  from  36  we  get 
nothing  for  a  remainder.  Our  quotient,  then,  is 
30  +  9,  or  39. 

Divide  432  by  6  ;  801   by  9 ;  336  by  4  ;  and 
525  by  7. 


56  Multiplication  and  Division.          [§  2. 

*12.  If  a  bunch  of  fire-crackers  costs  8  cents, 
how  many  bunches  can  you  buy  for  $1.28?  how 
many  bunches  for  |3.36  ? 

*13.  If  a  pound  of  brown  sugar  costs  9  cents, 
how  many  pounds  can  you  buy  for  $2.34  ?  how 
many  pounds  for  |9.36  ? 

*14.  A  man  bought  8  acres  of  land  for  $912; 
what  was  that  an  acre  ? 

*15.  How  many  miles  a  day  must  an  Atlantic 
steamer  make,  if  she  is  to  cross  the  ocean,  a  dis- 
tance of  2870  miles,  in  7  days?  how  many  miles  a 
day  if  she  is  to  cross  in  10  days  ? 

*16.  If  4  quarts  make  a  gallon,  how  many  gal- 
lons are  there  in  376  quarts  ?  how  many  gallons 
in  916  quarts  ? 

*17.  Sarah  wants  very  much  a  fine  writing-desk 
for  which  the  dealer  asks  $3.87.  If  she  gets  9 
cents  a  quart  for  picking  berries,  how  many  quarts 
must  she  pick  in  order  to  earn  enough  to  buy  the 
desk? 

*18.  Divide  792  by  6. 


1 


3 


2 


6)792  100         We   may  shorten  our  6)792 

600    30     work   by  omitting  some  6 

192             °^  the  figures,  thus :  jg 

180  18_ 

12  12 

12  12 

0 

Answer :     132. 

We  may  also  write  all  the  figures  of  our  quo- 
tient on  one  line,  as  in  the  following : 


C.]  Division.  57 


Divide  833  by  7. 
7)833(119. 

7 

13 

7 


Divide  968  by  4 

4)968(242.* 
8 

16" 
16 


63  8 

63  8 

Divide  855  by  9 ;  784  by  8 ;  and  795  by  5. 

*19.  If  a  barrel  of  flour  costs  8  dollars,  how 
many  barrels  can  be  bought  for  184  dollars? 

*2O.  The  4.30  express  from  Boston  to  New 
York  makes  the  distance  of  234  miles  in  6  hours ; 
how  many  miles  an  hour  must  the  train  cover? 

*21.  How  many  yards  of  cotton  cloth  at  9  cents 
a  yard  can  be  bought  for  I486  ? 

*22.  George  gets  9  cents  an  hour  for  weeding 
onions;  how  long  will  it  take  him  to  earn  $1.35? 

*23.  Where  the  divisor  is  a  small  number,  it  is 
usual  to  do  in  one's  head  a  large  part  of  the  work 
of  dividing. 

a.  Divide  725  by  5. 
5)725  \      5  into  7  goes  once  with  a  re- 

>  mainder  2 ;  we  set  down  the  1 
145  =  quotient.  \  Al  -  ,. 

;  under  the  7  and  put  the  re- 
mainder with  the  next  figure  of  the  dividend,  mak- 
ing 22.  5  into  22  goes  4  times,  with  a  remainder 
2  ;  we  set  down  the  4  as  the  second  figure  of  the 
quotient,  and  put  the  remainder  with  the  next  fig- 
ure of  the  dividend,  making  25.  5  into  25  goes  5 
times,  with  no  remainder.  5,  then,  is  the  last  fig- 
ure of  the  quotient,  and  725 -r  5  =  145. 


58  Multiplication  and  Division.  [§  2. 


b.  Show  that  595-^-5  =  119;  552-.-4  = 
*24.  a.  Divide  632  by  4. 

4)632   )      4  into  6  goes  once  with  2  over.     4  into 
>   23  goes  5  times  with  3  over.     4  into  32 
'   goes  8  times  with  no  remainder. 
6.  Divide  185  by  5. 
5)185  }      5  into  1  will  not  go;  5  into  18  goes 

(   3  times  with  3  over.     5  into  35  goes  7 
o7    \    ,. 
/    times. 

c.  Show  that  the  work  is  correct  in  each  of  the 
following  examples  : 

6)192    7)651     4)1264     5)7285     9)374504508 

32        ~93  316         1457  41611612 

*25.  The  distance  from  London  to  Edinburgh 
by  the  Great  Northern  Railway  and  its  connec- 
tions is  405  miles  ;  how  many  miles  an  hour  must 
the  daily  express,  which  covers  the  distance  in  9 
hours,  make? 

*26.  Where  the  divisor  is  large,  it  is  necessary 
to  write  out  in  full  most  of  the  work. 

Divide  496  by  16.  Divide  322  by  14. 

16)496(31.  14)322(23. 

48  28 

~~16  ~42 

16  42 

NOTE.  Where  the  successive  steps  are  set  down,  as  in  *26,  the 
operation  is  called  long  division;  where  only  the  result  is  set 
down,  as  in  *23  and  *24,  the  operation  is  called  short  division. 

*27.  Sometimes  we  can  tell  only  by  trial  what 
the  different  figures  of  the  quotient  are. 
Divide  2001  by  29. 


C.]  Division.  59 


29)2001(69. 
174 


261 
261 


Here,  in  order  to  get  the  first 
figure  of  the  quotient,  we  have  to 
find,  by  trial,  how  many  times  29 
is  contained  in  200. 


If  we  try  7  we  find  that  7  x  29 
=  203  ;  therefore  7  is  too  large  and  we  try  6.  In 
the  same  way  we  have  to  find  Ity  trial  how  many 
times  29  is  contained  in  261. 

Show  that  1862  -r  38  =  49,  and  that  4446  -r  78  =  57. 

28.  How  long  will  it  take  a  train  of  cars  which 
goes  at  a  rate  of  29  miles  an  hour  to  go  2842 
miles  ? 

29.  I  bought  127  bushels  of  potatoes  for  $95.25 ; 
how  much  was  that  a  bushel  ? 

30.  There  are  12  inches  in  a  foot,  and  63360 
inches  in  a  mile ;  how  many  feet  are  there  in  a 
mile  ? 

31.  How  many  barrels  of  flour  at  15.75  a  bar- 
rel can  I  buy  for  $86.25  ? 

32.  If  29694  dollars  were  to  be  equally  divided 
among  202  soldiers,  how  many  dollars  would  each 
receive  ? 

33.  A  man  bought  a  house  for  $1275,  which  he 
paid  for  at  the  rate  of  $15  a  month ;  how  long 
was  he  in  paying  for  the  house  ? 

34.  The  expenses  of  a  picnic  party  of  16  men 
and  14  ladies  were  $1.60  each.     The  men  paid  all 
the  expenses.     How  much  did  each  man  pay  ? 

35.  I  bought  13  chests  of  tea,  each  chest  con- 
taining 18  Ibs.,  for  $152.10.     How  much  was  the 
tea  per  pound  ? 


60  Multiplication  and  Division.  [§  2. 

36.  I  bought  173  barrels  of  sugar  for   13633, 
and  sold  it  for  $3935.75.     How  much  did  I  make 
on  each  barrel? 

37.  The  circumference  of  each  fore  wheel  of  a 
carriage  is  8  feet,  and  of  each  hind  wheel  12  feet ; 
how  many  more  turns  will  the  fore  wheels  make 
than  the  hind  wheels  in  going  5280  feet  or  1  mile? 
How  many  more  turns  in  going  from  Providence 
to  Worcester,  a  distance  of  44  miles  ? 

38.  The    number  of  square  miles  in  Siberia  is 
5,493,629,  and  in  Great  Britain  is  120,832.     How 
many  countries  the  size  of  Great  Britain  could  be 
made  out  of  Siberia,  and  how  many  square  miles 
would  be  left  over  ? 

39.  How  many  countries  the  size  of  Great  Brit- 
ain could  be  made  out  of  European  Russia  (2,261,- 
657   square  miles),  and  how  many  square  miles 
would  be  left  over  ? 

40.  How  many  countries  the  size  of  Great  Brit- 
tain  could  be  made  out  of  China  and  its  depend- 
encies (3,924,627  square  miles),  and  how  many 
square  miles  would  be  left  over  ? 

41.  How   many   countries    the    size    of   Japan 
(156,644    square   miles)    could    be   made    out   of 
China  and  its  dependencies,  and  how  many  square 
miles  would  be  left  over  ? 

[For  the  sizes  of  the  states  referred  to  in  the  next  five  exam- 
ples, see  pages  27-29.  ] 

42.  Into  how  many  states  the  size  of  Connect^ 
cut  could  Texas  be  divided,  and  how  many  square 
miles  would  be  left  over  ? 


C.]  Division.  61 

43.  Into  how  many  states  the  size  of  New  York 
could  California  be  divided,  and  how  many  square 
miles  would  be  left  over? 

44.  Into  how  many  states  the  size  of  New  Jer- 
sey could  New  York  be  divided,  and  how  many 
square  miles  would  be  left  over  ? 

45.  How   many  times    is   the    smallest   of  the 
United  States  contained   in  the  largest,  and  with 
what  remainder? 

46.  The  number  of   square  miles  in  Denmark 
is  14,553.     How  many  countries  the  size  of  Den- 
mark could  be  made  out  of  Maine?  how  many  out 
of   North    Carolina  ?    Georgia  ?   Mississippi  ?   and 
how  many  miles  in  each  case  woidd  there  be  left 
over  ? 

47.  A  man  took  a  3  days'  walking  journey;  on 
the  first  day  he  walked  16  miles,  on  the  second  day 
25  miles,  and  on  the  third  day  19  miles.     How 
many  miles  a  day  would  he  have  walked  if  he  had 
made  the  journey  in  the  same  number  of  days  but 
had  gone  the  same  distance  each  day?     Ans.  20 
miles  a  day,  which  is  called  the  man's  AVERAGE 
rate  for  the  3  days  that  he  walked. 

48.  A  man  walked  for  five  days,  going  16  miles 
the  first  day,  19  miles  the  second  day,  23  miles  the 
third  day,  21  miles  the  fourth  day,  and  26  miles 
the  fifth  day.     What  was  his  average  rate  per  day  ? 

[Get  J  of  the  entire  distance.] 

49.  At  the  end  of  a  fourteen  days'  journey  a 
man  found  that  he  had  spent  $49  for  traveling  ex- 
penses.    What  was  his  average  expense  per  day  ? 


62  Multiplication  and  Division.         [§  2. 

at  the  same  average  how  much  would   he  need  for 
a  journey  of  23  days  ? 

50.  The  following  figures  show  how  many  pupils 
attended  the  Planktown  Grammar  School  during 
the  first  five  days  of  the  first  week  in  April : 

Monday  38  ;  Tuesday  47  ;  Wednesday  47 ;  Thurs- 
day 42  ;  Friday  46. 

What  was  the  average  attendance? 

51.  What  is  the  average  of  the  numbers  1,  2,  3, 
4,5,6,7,8,  9?  t 

D.  Tables  nml  Questions  for  Practice. 

MULTIPLICATION   TABLE   TO    12   TIMES    12. 


1 

2 

3 

^4 
8 

5 

6 

7 
14 

8 

9 

10 

11 

12 

2 

4 

6 

10 
15 

12 
18 

16 
24 

18 

27 

20 

22 

24 

3 

6 

9 

12 

21 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

77 

72 
84 

7 

14 

21 

28 

35 

42 

.49 

56 

63 

70 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 
60 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

72 

84 

96 

108 

120 

132 

144 

To  find  how  many  3  times  9  are,  put  your  finger  on  the  3  in  the 


D.]        Tables  and  Questions  for  Practice.         63 

left-hand  column ;  then  move  the  finger  to  the  right  till  it  comes 
under  the  9  in  the  top  line,  and  there  you  find  27,  which  is  3  times 
9.  In  the  same  way  the  product  of  any  two  of  the  first  twelve 
numbers  may  be  found. 

1.  Repeat  the  multiplication  table  for  each  num- 
ber from  2  to  12  inclusive. 

2.  The  product  of  any  number  multiplied  by 
itself  is  called  its  SQUARE.     Thus  2  times  2,  or  4,  is 
called  the  square  of  2  ;  3  times  3,  or  9,  the  square 
of  3;  12  times  12,  or  144,  the  square  of  12. 

,  What  is  the  square  of  4  ?  of  5  ?  of  6  ?  of  7  ? 
of  8?  of  9?  of  10?  of  11? 

3.  Construct  a  table  which  shall  give  the  square 
of  each  number  from  13  to  25  inclusive. 

4.  Extend  the  following  multiplication  table  to 
10  times  20. 


1 

13 

14 

15 

16 

17 

18 

19 

20 

2 

26 

3 

39 

4 

52 

5 
6 

65 





78 

7 

91 

8 

104 

9 

117 

10 

130 

5.  Repeat  the    multiplication    table    both   for- 
wards and  backwards  from  once  13  to  10  times  13 ; 


64 


Multiplication  and  Division. 


proceed  in  like  manner  with  each  number  from  14 
to  20,  inclusive. 

NOTE.  A  multiplication  table  beyond  12  times  12  is  not  usually 
learned  in  school ;  it  is  believed,  however,  that  the  pupil  will  be 
amply  repaid  in  the  future  for  the  time  spent  in  learning  this  table 
now. 

6.  Divide  each  number  in  the  following  table  by 
each  number  less  than  20  that  is  contained  in  it 
without  a  remainder.  Thus  12  -r  2  =  6  ;  12  -r  3  =  4 ; 
12^4  =  3;  12-^6  =  2,  etc. 


12 

20 

26 

33 

39 

46 

52 

60 

66 

74 

80 

88 

96 

14 

21 

27 

34 

40 

48 

54 

62 

68 

75 

81 

90 

98 

15 

22 

28 

35 

42 

49 

55 

63 

69 

76 

82 

92 

99 

16 

24 

30 

36 

44 

50 

56 

64 

70 

77 

84 

94 

100 

18 

25 

32 

38 

45 

51 

58 

65 

72 

78 

86 

95 

7.  Each  of  the  following  numbers  is  the  square 
of  what  number  ? 

25,  36,  16,  49,  9,  64,  4,  81,  144,  169,  100,  196, 
121,  625,  225,  484,  256,  529,  324,  576,  441,  361, 
400,  289. 

8.  Multiply  the  first  number  in  column  B  (see 
p.  26)  by  the  first  number  in    column    C  ;    the 
second  number  in  column  B  by  the  second  num- 
ber in  column  C,  and  so  on  to  the  ends  of  these 
columns. 

9.  Divide  the  first  number  in  column  E   (see 
p.  26)  by  the    first    number    in  column    D ;    the 
second  number  in  column   E  by  the  second  num 
ber  in  column  D,  and  so  on  to  the  ends  of  these 
columns. 


SECTION  III. 
FRACTIONS. 

[This  section,  to  Example  53,  page  95,  is  drawn  from  the  Re- 
vised Edition  of  Colburn's  First  Lessons.  Most  pupils  will  do 
well  to  review  Fractions  carefully  as  a  preparation  for  the  study 
of  Decimals.  In  cases  where  a  review  is  not  deemed  necessary, 
pages  65-95  may  be  omitted.] 

A.  Fractional  Notation  and  fractional  Terms. 

Halves,  thirds,  fourths,  fifths,  etc.,  of  a  thing 
are  called  FRACTIONS. 

Fractions  may  be  expressed  by  figures,  thus : 

One  third  by  £.  Three  fourths  by  J. 

Two  thirds  by  §.         Two  fifths  by  f . 

One  fourth  by  J.         Four  sevenths  by  %. 

1.  Eead  the  fractions :  £,  J,  f ,  ^  &,  fa  W,  -ft, 

*,  *,  f  • 

2.  Express  one  fifth  by  figures  ;  three  fifths  ;  one 
sixth  ;    four  sixths  ;    five   sixths ;    three   sevenths ; 
one  eighth  ;  five  eighths  ;  one  ninth  ;  four  ninths  ; 
one  thirteenth ;    eleven    thirteenths ;    one  twenty- 
first  ;  twenty  twenty-firsts, 

3.  What  is  §  of  6  ?  J  of  8  ?  f  of  15  ? 

4.  What  is  f  of  14?  }  of  32?  §  of  18? 

5.  14  is  |  of  what  number? 

6.  18  is  ^  of  what  number? 


66  Fractions.  [§  3. 

7.  25  is  $  of  what  number? 

8.  If  I  plant  J  of  my  garden  with  peas,  \  with 
beans,  J  with  potatoes,  and  i  with  tomatoes,  how 
many  eighths  shall  I  have  left  for  other  vegetables  ? 

How  many  eighths  are  l+J  +  i  +  i? 
Add  together  f ,  J,  and  $. 

9.  How  many  sevenths  are  }  4-  ?  4-  ?  ? 

10.  How  many  ninths  are  $  +  §  +  }? 

As  we  have  seen,  it  requires  two  numbers  to  ex- 
press a  fraction ;  one  (which  we  write  below  the 
line)  to  show  into  how  many  equal  parts  the  thing 
which  we  are  talking  about  is  divided ;  the  other 
to  show  how  many  of  these  parts  are  taken. 

For  instance,  ?  of  an  apple  is  that  portion  of  an 
apple  which  we  get  by  cutting  the  apple  into  7 
equal  parts,  and  then  taking  3  of  these  parts.  W  °f 
an  orange  represents  that  portion  of  an  orange 
which  we  could  get  by  cutting  the  orange  into  12 
equal  parts,  and  taking  5  of  these  parts. 

11.  What  is  meant  by  |  of  an  orange  ?  J  of  an 
apple  ?  T7?  of  a  melon  ? 

The  number  which  shows  into  how  many  parts 
the  thing  is  to  be  divided  is  always  written  below 
the  line,  and  is  called  the  DENOMINATOR  of  the 
fraction.  The  number  which  shows  how  many  of 
these  equal  parts  are  to  be  taken  is  written  above 
the  line,  and  is  called  the  NUMERATOR.  Thus, 
in  J,  9  is  the  denominator  and  5  is  the  numerator* 


A.]  Fractional  Notation.  67 

12.  a.  In  the  fraction  J,  what  does  the  8  denote  ? 
the  7  ?  which  is  the  numerator  ?  which  the  denom- 
inator ? 

6.  Which  is  the  numerator  and  which  is  the  de- 
nominator in  each  of  the  following  fractions?  -fa,  f, 

T3TT>  §,  if- 

Notice  that  when  a  thing  is  divided  into  3  equal 
parts,  the  parts  are  called  thirds  ;  when  a  thing  is 
divided  into  4  equal  parts,  the  parts  are  called 
fourths;  when  a  thing  is  divided  into  19  equal 
parts,  the  parts  are  called  nineteenths  ;  that  is,  the 
fraction  takes  its  name  from  the  denominator. 


13.  Read  A,  A,  A,  A,  jf,  Jif, 

14.  Mary  had  }  of  a  melon,  but  gave  away  \  ; 
how  much  had  she  left? 

Whatisf-i?  ft-|?  «-*?  A-  A? 

15.  Mr.  Wood  bought  a  yacht,  and  sold  \  of  it 
to  Mr.   Shellabarger  ;  what  portion  of  the  yacht 
did  Mr.  Wood  then  own  ? 

16.  How  much  is 
A+A-A+H? 

Fractions  like  J,  §-,  J,  J,  £,  in  which  the  numera- 
tor is  smaller  than  the  denominator,  are  called 
PROPER  fractions. 

Fractions  like  f  ,  J,  J,  f  ,  in  which  the  numerator 
is  larger  than  the  denominator,  are  called  IMPROPER 
fractions. 

17.  Which  of  the  following  are  proper  fractions 
and  which   improper  fractions?    |,  f,  $,  J,  5,  A? 


68 


Fractions. 


[§3. 


18.  Change  f  to  a  whole  number  and  a  fraction. 
Answer :  f^f  +  i^l  +  J,  and  this  is  usually  writ- 
ten 1J. 

19.  Change  ^  to  a  whole  number  and  a  fraction. 

Answer :  2  and  f  ;  usually  written  2  J. 

20.  A  whole  number  and  a  fraction  like  li,  2|, 
4J,  is  called  a  MIXED  NUMBER. 

Change  the  mixed  number  4 ^  to  an  improper 
fraction.  Answer :  4i  =  4  +  £  =  *£  +  J  =  J33. 

21.  Change  the  mixed  numbers  3£,  If,  2J,  4J, 
3$  to  improper  fractions. 

22.  Change  the  improper  fractions  $,  J,  |,  g,  V, 
^,  to  mixed  numbers. 

B.   Common  Denominator.    Problems.     Illus- 
trations. 

1.  If  you  divide  an  apple  into  two  equal  parts, 
and  then  divide  each  one  of 
these  into  two  equal  parts,  how 
many  pieces  will  you  have? 
What  part  of  the  whole  apple 
will  each  piece  be  ? 

Answer  :    There    will    be   4 
equal  pieces,  so  that  each  piece 
will  be  one  fourth  of  the  apple. 
J  of  an  apple  is  equal  to  how 
many  fourths  of  an  apple  ? 

2.  How  many  fourths  of  an 
orange  are  there  in  J  of  an  orange  ? 
Answer :  2  fourths  of  an  orange. 
3.  If  you  give  J  of  an  orange  to  one  boy  and  J 


B.]  Common  Denominator.  69 

to  another,  how  much  more  do  you  give  the  first 
than  the  second  ? 

4.  If  you  give  £  of  an  orange  to  one  boy  and  J 
to  another,  how  many  times  £  of  an  orange  do  you 
give  away?  how  many  fourths  of  the  orange  do  you 
have  left  ? 

How  many  times  J  are  J  and  J  ? 

5.  A  man  gave  to  one  laborer  \  of  a  bushel  of 
wheat,  and  to  another  J  of  a  bushel ;  how  many 
times  J  of  a  bushel  did  he  give  to  both  ?  how  many 
bushels  ?     How  many  times  \  are  \  and  f  ? 

6.  If  you  divide  an  inch  into  halves,  and  then 
divide  each  of  these  halves  into  ^    ^ 

3  equal  parts,  how  many  parts 

will  you  have  ?  what  portion  of  an  inch  will  each 

part  be  ? 

How  many  sixths  of  an  inch  are  there  in  £  of 
an  inch  ? 

7.  A  man  gave  i  of  a  barrel  of  flour  to  one 
person,  and  §  of  a  barrel  to  another ;  to  which  did 
he  give  the  more  ? 

8.  A  man,  paying  some  money  to  his  laborers, 
gave  each  man  \  of  a  dollar,  and  each  boy  i  of  a 
dollar  ;  how  much  more  did  he  give  to  a  man  than 
to  a  boy  ?  how  much  did  a  man  and  a  boy  receive  ? 

9.  What  is  the  difference  between  J  and   \  ? 
what  is  the  sum  of  J  and  J  ? 

10.  If  a  man  earns  *  of  a  dollar  in  a  day,  and  a 
boy  \  of  a  dollar,  how  much  more  does  the  man 
earn  than  the  boy  ? 

1 1 .  What  is  the  difference  between  f  and  \  ? 


70 


Fractions. 


[§3. 


12.  If  you  cut  a  loaf  of  cake  into  two  equal 
parts,  and  then   cut  each  part 

into  quarters,  how  many  pieces 
will  you  have  in  all  ?  what  part 
of  the  whole  loaf  will  each 
piece  be  ? 

13.  How  many  eighths  of  a  loaf  are  there  in  \ 
of  a  loaf? 

In  i  of  anything  there  are  how  many  eighths  ? 

14.  If  you  cut  a  line  or  loaf  of  cake  into  quar- 
ters, and  then  cut  each  one  of  these  quarters  into 
thirds,  how  many  pieces  will  you  have  in  all  ?  what 
part  of  the  whole  will  each  piece  be?  into  how 
many  twelfths  of  a  cake  can  you  cut  one  fourth  of 
a  cake  ? 

15.  Into  how  many  eighths  of  an  apple  can  you 
cut  $  of  an  apple? 

How  many  eighths  are  there  in  J  ? 

16.  If  you  cut  one  quarter  of  a  sheet  of  paper 
into  4  equal  parts,  what  portion  of  the  whole  sheet 
will  each  part  be  ? 

[The  pupil  should  find  out  the  answer  by  actually  cutting  a 
sheet  of  paper.] 

If  one  cake  of  chocolate  will  make  16  cups,  how 
many  cups  will  \  of  a  cake  make  ? 
How  many  sixteenths  are  there  in  J  ? 

17.  This  figure  is  supposed  to  represent  a  sheet 

of    paper    divided     into    equal 
pieces.     What  part  of  the  whole 
sheet  is  one  of  these  pieces  ? 
2   pieces   form   what  part   of 


B.]  Common  Denominator.  71 

the  whole  sheet  ?     How  many  twelfths  are  there  in 
one  sixth  ? 

3  pieces  form  what  part  of  the  whole  sheet?  9 
pieces  ?     How   many   twelfths   are   there   in   one 
fourth  ?  how  many  in  3  fourths  ? 

4  pieces  form  what  part  of  the  whole  sheet  ?  8 
pieces  ?     How   many   twelfths   are   there   in   one 
third  ?  how  many  in  2  thirds  ? 

6  pieces  form  what  part  of  the  whole  sheet  ? 
How  many  twelfths  are  there  in  one  half  ?  fy 

18.  Take  a  sheet  of  paper  six  inches  long  and  4 
inches  wide,  and  draw  a  line  on  it  so  as  to  divide  it 
into  halves  ;  now  draw  a  second  line  so  as  to  divide 
the  sheet  into  quarters  ;  now  draw  4  other  lines, 
so  as  to  divide  the  sheet  into  twelfths. 

Cut  out  with  the  scissors  one  twelfth  of  the 
sheet  ;  also  one  sixth.  How  long  and  how  wide 
is  each  of  these  pieces  ? 

Cut  out  one  fourth  of  the  sheet  ;  also  one  third. 
How  long  and  how  wide  is  each  of  these  pieces  ? 

How  many  twelfths  of  the  sheet  have  you  cut 
out  in  all  ?  How  many  twelfths,  then,  are 


19.  How  many  twelfths  are  equal  to  £  ? 

We  may  ask  ourselves  into  how  many  twelfths 
of  anything  (for  instance,  a  sheet  of  paper  or  a 
cake)  we  can  cut  \  of  the  thing,  or  we  may  reason 
thus: 

In  anything  there  are  12  twelfths  of  that  thing  ; 
in  j-  of  the  thing,  then,  there  must  be  \  of  12 
twelfths,  or  4  twelfths.  Therefore  £  =  iV 


72  Fractions.  [§  3. 

20.  \  is  equal  to  how  many  tenths  ? 

Answer :  In  1  there  are  10  tenths  ;  in  J  of  1, 
then,  there  must  be  \  of  10  tenths,  or  5  tenths. 
Therefore  $  =  &. 

21.  I  is  equal  to  how  many  thirtieths  ? 
Answer :  In  1  there  are  30  thirtieths  ;  in  J  of 

1,   then,  there   must  be  £  of    30   thirtieths,  or  G 
thirtieths.     Therefore  J^^a- 

22.  \   is  equal  to  how   many  twelfths  ?    four- 
teenths ?  twenty-fourths  ? 

23.  ^  is  equal  to  how  many  ninths  ?  fifteenths  ? 
twenty-fourths  ? 

24.  £  is  equal  to  how  many  twentieths  ?  twenty- 
fourths  ?  thirty-seconds  ? 

25.  }  is  equal  to  how  many  fourteenths?  twenty- 
eighths  ? 

26.  How  many  thirty-sixths   does    it    take    to 
makei?  i?  1?  i?  i?  T»5?  &  ? 

27.  How  many  sixtieths  does  it  take  to  make 

A?  A?  A?  t?  t?  i?  4?  i? 

28.  A  man  bought  J  of  a  bushel  of  wheat  at  one 
time,  and  f  of  a  bushel  at  another ;  at  which  time 
did  he  buy  the  more  ? 

29.  An  ounce  of   Mary's   medicine  makes  48 
closes  ;  what  part  of  an  ounce  is  there  in  a  dose  ? 
how  many  doses   are   there   in   a  quarter  of   an 
ounce?     i  is  equal  to  how  many  forty-eighths  ? 

30.  §  are  equal  to  how  many  ninths  ? 
Answer :   One  third  equals  3  ninths,  therefore 

two  thirds  must  equal  2  times  3  ninths  or  6  ninths. 

§—  « 
—  ¥• 


B.]  Common  Denominator.  73 

31.  ^  are  equal  to  how  many  twenty -eighths  ? 
Answer :   One  seventh  equals  4  twenty-eighths, 

therefore   five   sevenths    must    equal    5    times   4 
twenty-eighths  or  20  twenty-eighths. 

32.  Show  that  $  =  i$;  |  =  ft;  A  =  A;  f  =  H- 

33.  tf  are  how  many  times  ft  ? 
f  are  how  many  times  £T  ? 
f  are  how  many  times  ^  ? 

34.  |  are  how  many  times  ^  ? 
f  are  how  many  times  3^  ? 
ft  are  how  many  times  ^  ? 

35.  How  many  twelfths  are  there  in  §  ?  in  |  ? 
in  J? 

36.  How  many  eighteenths  are  there  in  J?  in  J? 
in  I? 

37.  How  many  forty-eighths  are  there  in  §  ?  in 
|?  in£?  in§?  in  A?  in  ft?  in  JJ  ? 

38.  A  man  bought  §  of  a  yard  of  cloth  at  one 
time,  and  g  of  a  yard  at  another  ;  how  many  sixths 
of  a  yard  did  he  buy  altogether  ?  at  which  time 
did  he  buy  the  more? 

39.  What   shall    we   get   if  we    multiply  both 
numerator  and  denominator  of  \  by  2  ? 

First:  If  we  multiply  the  denominator  by  2 
we  get 

1        1 
2x2  "  4 

Second :  If  we  now  multiply  the  numerator  1 
also  by  2  we  get 

2x1  _2 

^~=4~ 


74  Fractions.  [§  3. 

But  |  is  the  same  as  J,  which  is  the  thing  we 
started  with.  Therefore  if  we  multiply  both  the 
numerator  and  the  denominator  of  J  by  2,  we  do 
not  alter  the  value  of  the  fraction. 

Shall  we  alter  the  value  of  the  fraction  \  if  we 
multiply  both  numerator  and  denominator  by  3  ? 
by  4? 

40.  What  shall  we  get  if  we  multiply  both 
numerator  and  denominator  of  the  fraction  J  by  4  ? 

Suppose  our  J  to  be  J  of  a  foot  (12  inches). 
J  of  a  foot  =  4  inches. 

If  we  multiply  the  denominator  of  \  by  4  we  get 

1     or.1 


4x3       12 
•fv  of  a  foot=  1  inch. 

If  we  now  multiply  the  numerator  by  4  we 
get  A. 

i4*  of  a  foot  =  4  inches. 

But  4  inches  is  what  we  started  with  ;  therefore 
we  have  not  altered  the  value  of  J  by  multiplying 
both  its  numerator  and  its  denominator  by  4. 

Shall  we  alter  the  value  of  the  fraction  J  if  we 
multiply  both  numerator  and  denominator  by  2  ? 

41.  If  we  multiply  both  numerator  and  denom- 
inator of  f  by  5  we  shall  get  Jf  ;  let  us  compare 
this  result  with  f  : 

I  —  -fs  >  therefore  ^  =  3  times  -f$  or  ^f ,  and  we  see 
that  we  have  not  altered  the  value  of  the  fraction 
^  by  multiplying  both  numerator  and  denominator 
by  5. 

Show  that  if  you  multiply  both  numerator  and 


B.]  Common  Denominator.  75 

denominator  of  f  by  6  you  will  not  alter  the  value 
of  the  fraction  ;  of  /T  by  4  ;  of  if  by  2 ;  of  ^  by  6. 
In  the  same  way  you  can  take  any  fraction  and 
show  that  its  value  will  not  be  changed  if  you 
multiply  both  numerator  and  denominator  by  the 
same  number,  whatever  that  number  may  be. 

42.  The  numerator  and  denominator  of  a  frac- 
tion  are   sometimes  called  its  TERMS.     Thus  the 
terms  of   the  fraction  ^  are  3  and  7.     By  what 
must  you  multiply  both  terms  of  the  fraction  J  in 
order  to  get  an  equivalent  fraction  whose  denom- 
inator is  12  ? 

43.  By  what  must  you  multiply  both  terms  of 
the  fraction  £  in  order  to  change  it  to  eighths  ?  to 
sixteenths  ?  to  twenty-fourths  ? 

44.  Multiply  both  terms  of  each  of  the  following 
fractions  by  something  that  will  change  the  frac- 
tion to  sixty-fourths  :  1 ;  i  ;  f  ;  -ft. 

45.  Change  to  forty-eighths  J,  j,  j,  g,  J,  H, 

H,  H- 

46.  What  shall  we  get  if  we  divide  both  nu- 
merator and  denominator  of  ^  by  3  ? 

Suppose  our  T9^  to  be  ^  of  a  foot  (12  inches). 

•fs  of  a  foot  =  9  inches. 
If  we  divide  the  numerator  of  fy  by  3  we  get  ^. 

^  of  a  foot  =  3  inches. 

If  we  now  divide  the  denominator  of  -ft-  by  3  we 
get  f. 

|  of  a  foot  =  9  inches. 

But  9  inches  is  what  we  started  with  ;  therefore 
we  have  not  altered  the  value  of  -fy  by  dividing 
both  its  numerator  and  its  denominator  by  3. 


76  Fractions  [§  3. 

What  will  you  get  if  you  divide  both  numerator 
and  denominator  of  ^2  by  4  ? 

47.  Divide    both   numerator    and    denominator 
of  }g  by  3  and  compare  the  result  with  \$. 

12-r3_4 
15-3    5* 

J=  ?5 ;  therefore  $  =  4  times  ^  or  JJ:  we  have, 
then,  not  altered  the  value  of  \l  by  dividing  both 
numerator  and  denominator  by  3. 

Show  that  if  you  divide  both  numerator  and 
denominator  of  JJ  by  4  you  will  not  alter  the  value 
of  the  fraction  ;  of  |-g  by  10  ;  of  if  by  9  ;  of  }$  by 
12  ;  of  ft  by  7. 

In  the  same  way  you  can  take  any  fraction  and 
show  that  its  value  will  not  be  changed  by  dividing 
both  numerator  and  denominator  by  the  same 
number. 

48.  When,  as  in  the  last  question,  we  divide 
both  the  numerator  and  denominator  by  the  same 
number,  we  are  said  to  reduce  the  fraction  to  lower 
terms ;  and  when  there  is  no  number  that  will  go 
in  both  the  numerator  and  denominator  without  a 
remainder,  the  fraction  is  said  to  have  been  reduced 
to  its  lowest  terms. 

If,  for  example,  we  divide  both  terms  of  -ft-  by 
2,  we  reduce  it  to  lower  terms  and  get  f  ;  if  we 
now  divide  both  terms  of  f  by  2  we  get  §,  and  have 
reduced  our  fraction  T9^  to  its  lowest  terms,  for 
there  is  no  number  (except  1)  which  will  go  in 
both  2  and  3  without  a  remainder. 

Reduce  J$  to  its  lowest  terms.  Answer :  f . 


B.]  Common  Denominator.  17 

49.  Reduce  to  lowest  terms  the  fractions  f,  y5^, 

I,  A>  «,  A,  A,  M>  H,  A>  M,  f  i>  H>  M>  M,  1!,  M, 
fi,  **,  H,  and  f  5. 

50.  Where  the  denominators  of  two  fractions 
are  the  same,  the  fractions  are  said  to  have  a  com- 
mon denominator.    Thus,  J  and  §  have  the  common 
denominator  8. 

When  two  fractions  have  a  common  denominator 
we  may  add  them  together  by  adding  the  numer- 
ators and  writing  the  sum  over  the  common  de- 
nominator, thus  : 

4 ,3_ 4+3_7 
9  +  9~~9 9' 

When  we  have  to  add  together  two  fractions 
which  have  different  denominators  we  may  first 
change  the  fractions  into  equivalent  ones  which 
shall  have  a  common  denominator,  and  then  add 
them,  thus  :  §  +  f  =  A  +  ft  =  H- 

J  is  how  many  twelfths  ?  £  is  how  many  twelfths  ? 
I  and  %  are  how  many  twelfths  ? 

51.  J  is  how  many  twentieths?  £  is  how  many 
twentieths  ?  J  and  J  are  how  many  twentieths  ? 

52.  §   are   how   many   fifteenths  ?    f   are  how 
many  fifteenths  ?  §  and  J  are  how  many  fifteenths  ? 

53.  §  are  how  many  sixths  ?  §  less  £  are  how 
many  sixths  ? 

54.  Change   £  and  §  to  eighteenths  and  then 
add  them. 

Answer :  J  =  ft  ;  f  =  TV     Therefore  J  +  f  =  T|. 

55.  Change  ^  and  j  to  thirty-fifths  and  then 
add  them.     Which  is  the  larger,  and  how  much  ? 


78  Fractions.  [§  3. 

56.  Change  f  and  §  to  forty-fifths  and  then  add 
them.     Which  is  the  larger,  and  how  much? 

57.  A  man  had  a  horse,  a  cow,  and  a  sheep. 
The  horse  eat  §  of  a  load  of  hay  in  the  winter,  the 
cow  J,  and  the  sheep  £.     How  many  sixths  of  a 
load  did  each  eat  ?  how  many  sixths  did  they  all 
eat? 

58.  A  boy,  having  a  quart  of  nuts,  wished  to 
divide  them  so  as  to  give  one  companion  J,  another 
J,  and  a  third  J  of  them  ;  but,  in  order  to  make  a 
proper  division,  he  first  separated  the  whole  into 
eight  equal  parts,  and  then  he  was  able  to  divide 
them  as    he  wished.     How  many  eighths  did   he 
give  to  each  ?  how  many  eighths  had  he  left  for 
himself  ? 

If  we  wish  to  replace  two  fractions  by  equivalent 
fractions  having  a  common  denominator  we  may 
choose  for  our  common  denominator  any  number 
which  contains  both  denominators  without  a  re- 
mainder. 

For  instance,  if  our  fractions  are  f  and  f  we 
may  choose  36  for  our  common  denominator ;  for 
36  contains  both  4  and  9  without  a  remainder. 

If  our  fractions  are  g  and  J  we  may  choose  12, 
or  24,  or  36,  or  48 ;  for  each  of  these  contains  both 
6  and  4  without  a  remainder,  but  it  is  convenient 
to  take  as  small  a  number  as  we  can,  so  in  this  case 
we  should  choose  12. 

We  can  always  find  a  number  which  will  contain 
both  of  our  denominators  without  a  remainder,  by 
multiplying  them  together. 


B.]  Common  Denominator.  79 

59.  Reduce  J  and  f  to  equivalent  fractions  hav- 
ing a  common  denominator. 

Solution :  "7  x  9  =  63  contains  both  7  and  9 
without  a  remainder,  and  may  be  taken  as  our 
common  denominator,  i  — /^5  ?  — 15- 

60.  Reduce  to  a  common  denominator  J  and  $ ; 
$  and  f  ;  |  and  §  ;  J.  and  f  ;  f  and  g. 

61.  Add  together  §  and  f ;  ^  and  ^?;  f  and  T\; 
£  and  §  and  y3(). 

62.  Subtract  J  from  J ;  ^  from  f  ;  f  from  § ; 
§  from  f . 

63.  A  boy,  distributing  some  nuts   among  his 
companions,  gave  J  of  a  quart  to  one,  and  ^  of  a 
quart  to  another;  how  much  more  did  he  give  to 
one  than  to  the  other? 

64.  What  is  the  difference  between  J  and  £  ? 

65.  A  man,  having  two  bushels  of  grain  to  dis- 
tribute among  his  laborers,  wished  to  give  J  of  a 
bushel  to  one,  and  §  of  a  bushel  to  another,  and  the 
rest  to  a  third,  but  was  at  a  loss  to  tell  how  to 
divide   it ;  at   last   he   concluded   to   divide   each 
bushel  into  six  equal  parts,  or  sixths,  and  then  to 
distribute  those  parts.     How  many  sixths  did  he 
give  to  each  ? 

66.  §  is  how  many  sixths  ? 

67.  A  man,  who  had  a  bushel  of  wheat,  wished 
to  give  ^  of  it  to  one  man,  and  J  of  it  to  another  ; 
but  he  could  not  tell  how  to  divide  it.     Another 
man  standing  by  advised  him  to  divide  the  whole 
bushel  into  six  equal  parts  first,  and  then  take  i  of 
them  for  one,  and  £  of  them  for  the  other.     How 


80  Fractions.  [§  3. 

many  parts  did  he  give  to  each?   how  many  to 
both  ?  how  many  had  he  left  ? 

68.  i    is   how  many   sixths?    £   is   how   many 
sixths  ?  i  and  ^  are  how  many  sixths  ? 

69.  £  and  J  are  how  many  times  J? 

70.  J  and  \  are  how  many  times  J  ? 

71.  |  and  i  are  how  many  times  J? 

72.  |  and  f  are  how  many  times  J  ? 

73.  J  and  §  are  how  many  times  J  ? 

74.  £  and  £  are  how  many  times  ^  ? 

75.  J  and  £  and  J  are  how  many  times  J? 

76.  i  and  f  and  ^  are  how  many  times  TV  ? 

77.  §  and  J  are  how  many  times  fa? 

78.  §  and  J  and  j  are  how  many  times  -fal 

79.  J  and  ^  and  J  are  how  many  times  ^  ? 

80.  \  and  J  and  J  and  J  and  ^  are  how  many 
times  3^  ? 

81.  J  and  ^  are  how  many  times  -fal 

82.  g  and  J  are  how  many  times 

83.  §  and  J-  are  how  many  times 

84.  §  less  £  are  how  many  times  J  ? 

85.  Which  is  the  larger,  }  or  f  ?   how  much  the 
larger  ? 

86.  A  boy,  having  a  pound  of  almonds,  said  he 
intended  to  give  J  of  them  to  his  sister,  and  J  to 
his  brother,  and  the  rest   to   his   mamma.      His 
mamma,  smiling,  said  she  did  not  think  he  could 
divide  them  so.     "  Oh,  yes,  I  can,"  said  he ;  "I 
will  first  divide  them  into  twelve  equal  parts,  and 
then  I  can  divide  them  well  enough."     Pray,  how 
many  twelfths  did  he  give  to  each  ? 


B.]  Common  Denominator.  81 

87.  i  is  how  many  times  T^?  J  is  how  many 
times   A  ?  £  and  \  are  how  many  times  -^  ? 

88.  Mr.  Goodman,  having  a  pound  of  raisins, 
said,  "  I  will  give  Sarah  J,  and  Mary  £ ,  and  James 
\  of  them,  and  the  rest  shall  go  to  Charles,  if  he 
can    tell   how   to   divide     them."      "  Well,"    said 
Charles,    "  I  would    first  divide    the    whole   into 
twelve  equal  parts,  and  then  I  could  take  \  and  \ 
and  £  of  them."     How  many  twelfths  would  each 
have? 

89.  J  and  \  and  \  are  how  many  times  TV  ? 

90.  George  bought  a   pine-apple,  and  said  he 
would  give  -]-  of  it  to  his  papa,  and  §  to  his  mamma, 
and  f$  to  his  brother  James,  if  James  could  di- 
vide it.     James  took  it,  and  cut  it  into  twenty 
equal  pieces,  and  then  distributed  them  as  George 
had  desired.     How  many  twentieths  did  he  give 
to  each  ? 

91.  §  and  f  less  /.,  are  how  many  times 

92.  g  less  $  are  how  many  times 

93.  #  less  J  are  how  many  times 

94.  f  less  f  are  how  many  times 

95.  i  and  f  and  J  and  ^  less  g,  are  how  many 
times  ^  ? 

96.  i  and  J  and  f  and   TV  and  ^V»  less  J,   are 
how  many  times  ^  ? 

97.  |  and  J  are  how  many  times 

98.  f  and  ^  are  how  many  times 

99.  ^  and  f  are  how  many  times  ^V? 

100.  Mr.  Fuller  said  he  would  give  £  of  a  pine- 
apple to  Fanny,  and  f  to  George,  and  the  rest  to 


82  Fractions.  [§  3. 

the  one  that  could  tell  how  to  divide  it,  and  how 
much  there  would  be  left.  But  neither  of  them 
could  tell ;  so  he  kept  it  himself.  Could  you  have 
told,  if  you  had  been  there?  How  would  you 
have  divided  it?  how  much  would  have  been  your 
share  ? 

101.  A  man  sold  1|  bushels  of  wheat  to  one 
man,  and  4|  to  another ;  how  many  did  he  sell  to 
both? 

102.  A  man  bought  6i  bushels  of  wheat  at  one 
time,  and  2J  at  another ;  how  much  did  he  buy 
in  all? 

103.  A  man  bought  7§   yards  of  one  kind  of 
cloth,  and  6J  of  another  kind ;  how  many  yards 
in  all? 

104.  A  man  bought  J  of  a  barrel  of  flour  at 
one  time,  and  2^  barrels  at  another,  and  6|  at  an- 
other ;  how  much  did  he  buy  in  all  ? 

105.  A  man  bought  one   sheep   for  4§  dollars, 
and  another  for  5$  dollars ;  how  much  did  he  give 
for  both? 

106.  There  is  a  pole  standing  so  that  J  of  it  is 
in  the  mud,  and  ?  of  it  in  the  water,  and  the  rest 
out  of  the  water ;  how  much  of  it  is  out  of  the 
water  ? 

107.  A  man,  having  undertaken  to  do  a  piece 
of  work,  did  £  of  it  the  first  day,  J  of  it  the  sec- 
ond day,  and  £  of  it  the  third  day ;  how  much  of 
it  did  he  do  in  three  days  ? 

108.  A  man  having  a  piece  of  work  to  do,  hired 
two  men  and  a  boy  to  do  it.     The  first  man  could 


C.]  Multiplication  of  Fractions.  83 

do  £  of  the  work  in  a  day,  and  the  other  J  of  it, 
and  the  boy  £  of  it ;  how  much  of  it  could  they 
all  do  in  a  day  ? 


C.  Multiplication  of  Fractions. 

1 .  A  boy,  having  £  of  an  apple,  gave  away  J  of 
what  he  had  ;  what  part  of  the  whole  apple  did  he 
give  away? 

Answer :  J  of  an  apple  is  the  same  as  f  of  an 
apple  therefore  the  boy  gave  away  J  of  £  of  an 
apple  or  \  of  an  apple. 

2.  What  is  £  of  J? 

3.  If  you  cut  an  orange  into  three  equal  pieces, 
and  then  cut  each  of  those  pieces  into  two  equal 
pieces,  how  many  pieces  will  the  whole  orange  be 
cut  into  ?     What  part  of  the    whole  orange  will 
one  of  the  pieces  be  ? 

4.  What  is  i  of  4  ? 

5.  A  boy  had  i  of  a  pine-apple,  and  cut  that 
half  into  three  equal  pieces,  in  order  to  give  away 
J  of  it.     What  part  of  the  whole  apple  did   he 
give  away? 

6.  What  is  4  of  J? 

7.  If  an  orange  is  cut  into  4  parts,   and   then 
each  of  the  parts  is  cut   in   two,  how  many  pieces 
will  the  whole  be  cut  into  ? 

8.  What  is  £  of  i  ? 

9.  A  man  having  £  of  a  barrel  of  flour,  sold  J  of 
what  he  had ;  how  much  did  he  sell? 


84  Fractions.  [§  3. 

10.  What  is  J  of  J  ? 

11.  If  an  orange  be  cut  into  4  equal  parts,  and 
each  of  these  parts  be  cut  into  3  equal  parts,  into 
how  many  parts  will  the  whole  orange  be  cut  ? 

12.  What  is  i  of  J? 

13.  A  boy,  having  £  of  a  quart  of  chestnuts, 
gave  away  J  of  what  he  had.     What  part  of  the 
whole  quart  did  he  give  away  ? 

14.  What  is  £  of  J  ? 

15.  What  is  £  of  £? 

16.  A  man,  who  owned  J  of  a  ship,  sold  J  of 
his  share ;  what  part  of  the  ship  did  he  sell,  and 
what  part  did  he  then  own  ? 

[When  we  get  J  of  J  we  are  said  to  multiply 
the  fractions  together  ;  we  can,  then,  use  the  mul- 
tiplication sign  ( x  )  instead  of  the  word  "  of," 
thus  £  x  J.] 

17.  Whatis  Jxj? 

18.  What  is  Jx  J? 

19.  What  is  Jx  J? 

20.  What  is  i  of  J? 

21.  What  is  J  of  j? 

22.  What  is  J  of  J? 

23.  What  is  £  of  J  ? 

24.  What  is  J  of  J  ? 

25.  A  boy,  having  §  of  an  orange  (that  is,  2 
pieces),  gave   his  sister   £  of  what  he  had  ;  how 
many  thirds  did  he  give  her  ? 

26.  What  is  i  of  §  ? 

27.  A   boy,   having  j  of  a  pine-apple,  said  he 
would  give  one  half  of  what  he  had  to  his  sister, 


C.]  Multiplication  of  Fractions.  85 

if  she  could  tell  how  to  divide  it.  His  sister  said, 
"  You  have  f ,  or  three  pieces,  if  you  cut  them  all 
in  two,  you  can  give  me  \  of  each  one  of  them." 
Flow  much  did  his  sister  receive  ? 

28.  What  is  \  of  |  ? 

29.  A  man,  who  owned  f  of  a  share  in  a  Boston 
bank,  sold  J  of  his  part ;  what  part  of  a  share  did 
he  sell  ? 

30.  What  is  J  of  |  ? 

31.  A  man,  who  owned  J  of  a  ship,  sold  J  of  his 
share  ;  what  part  of  the  whole  ship  did  he  sell  ? 
What  part  had  he  left  ? 

32.  What  is  J  of  f? 

33.  What  is  J  of  $? 

34.  What  is  i  of  ^  ? 

35.  What  is  i  of  ?? 

Answer :  \  of  \  =  ^ ,  therefore  J  of  ^  will  be  3 
times  as  much,  or  ^. 

36.  A  man,  who  owned  f  of  a  share  in  a  bank, 
sold  £  of  his  part  ;  what  part  of  a  whole  share  did 
he  sell  ? 

37.  What  is  J  of  £? 

38.  What  is  £  of  f  ? 

39.  A  boy,  having  J  of  a  watermelon,  wished  to 
divide  his  part  equally  among  his  sister,  his  brother, 
and  himself,  but  was  at  a  loss  to  know  how  to  do 
it ;  but  his  sister  advised  him  to  cut  each  of  the 
fifths  into   3  equal  parts.     How  many  pieces  did 
each  have  ?  and  what  part  of  the  whole  melon  was 

piece  ? 

40.  What  is     of  f  ? 


86  'Fractions.  [§  3. 

41.  What  is  J  of  f  ? 

42.  What  is  J  of  J? 

43.  What  is  J  of  i  ? 

44.  What  is  i  of  §? 

45.  What  is  f  of  §  ? 

Answer  :  We  first  get  J  of  £,  Now  *  of  J  =  5^ ; 
i  of  f  is,  then,  twice  as  much,  or  ^,  which  equals 
A-  Therefore  £  of  £ ,  which  is  3  times  J  of  £, 

is  A- 

46.  What  is  4  of  £  ? 

47.  What  is  §  of  f  ? 

48.  What  is  J  of  f  ? 

49.  What  is  §  of  4  ? 

50.  What  is  &  of  J? 

51.  What  is  Jof  ?? 

52.  What  is  J  of  f  ? 

53.  What  is  A  of  |  ? 

54.  What  is  A  of  |  ? 

55.  If  a  yard  of  cloth  costs  2  J  dollars,  what  will 
J  of  a  yard  cost  ? 

Answer :  If  1  yard  costs  $2^  or  f  £,  ^  a  yard 
will  cost  \  as  much  or  $|,  which  is  $lj. 

56.  What  is  J  of  2£? 

57.  A  boy  had  2^  oranges,  and  wished  to  give 
J  of  them  to  his   sister,  and  J  to  his  brother,  but 
he  did  not  know  how  to  divide  them  equally.     His 
brother  told  him  to  cut  the  whole  oranges   into 
halves,  and  then  cut  each  of  the  halves  into  3 
pieces.     What  part  of  a  whole  orange  did  each 
have  ? 

58.  What  is  4  of 


C.]  Multiplication  of  Fractions.  87 

59.  A  man  bought  4  bushels  of  corn  for  3§  dol- 
lars ;  what  part  of  a  dollar  did  1  bushel  cost  ? 

Answer  :  $3§  =  f^.     If  4  bushels  cost  $Y>  one 
bushel,  £  of  4  bushels,  will  cost  i  of  f  V  or  ii:  J- 

60.  What  is  £  of  3§  ? 

61.  If  5  bushels  of  wheat  cost  7f  dollars,  what 
is  that  a  bushel  ? 

62.  What  is  J  of  7f  ? 

63.  A  man  bought  6  gallons  of  alcohol  for  8f 
dollars  ;  what  was  that  a  gallon  ? 

64.  What  is  J  of  8f  ? 

65.  A  man  bought  7  bushels  of  potatoes  for  8^ 
dollars ;  how  much  was  that  a  bushel  ? 

66.  What  is  \  of  8f  ? 

67.  A  man   bought   10  pieces  of  nankeen  for 
6 1  dollars  ;  how  much  was  it  a  piece  ? 

68.  What  is  ^  of  6§  ? 

69.  If  9  bushels  of  rye  cost  6?  dollars,  what  is 
that  a  bushel  ? 

70.  What  is  Jx6£? 

71.  What  is  4x5?? 

72.  What  is  Jx8|? 

73.  What  is  i  x  6ft  ? 

74.  What  is  £  x  9*  ? 

75.  A  man  bought  7  yards  of  cloth  for  18|  dol- 
lars ;  what  was  that  a  yard  ?    What  would  3  yards 
cost? 

76.  What  is  \  of  18j  ?     What  is  f  of  18 J  ? 

77.  A  man  bought  5  barrels  of  vinegar  for  27 1 
dollars  ;  what  was  it  a  barrel  ?     What  would   7 
barrels  cost  at  that  rate  ? 


88 


Fractions. 


[§»• 


78.  What  is  J  of  27|  ?     What  is  J  of  27|  ? 

79.  If  6  barrels  of  flour  cost  38$  dollars,  what 
would  10  barrels  cost  at  that  rate  ? 

80.  What  is  -V  x  38|  ? 


81.  Show  that  ^x  _  =  . 

1       3         3        .  1    . 
5  =  3-irrr5and30 

Therefore  QXC==Q— z 
o     0     o  x  o 

82.  Show  that  s  x  w  =  - 


'15* 


15' 

x  f  will  be  4  times  as  much  as  J  x  J. 

^Xg  =  JL=A.    (See  Example  81.) 

4 
15' 


To  get  J  of  a  fraction,  that 
is,  to  multiply  it  by  £,  we 
may  multiply  its  denominator 


83.  Show  that 

V-  *  .!•.! 

3    4    3x4    12' 
lxl_    1    _1. 

3S/    /T 
X  O 

r 

32    3x2     6  ' 
12       2     _2  . 

> 

ma 

33    3x3     9  ' 
1V2      2        2 

by 

35    3x5    15' 

84.  Show  that 


14 


We  know  from  Example  82  that  ~  x  -  =  -  —  - 

o     o     o  x  o 


4 

™ 

J.D 


c.] 


Multiplication  of  Fractions. 


89 


Now  f  x  f  is  2  times  as  much  as  ^  x  ; 


^  '  2    4     2x4 

therefore  ~  x  -  =  - — -  = 
o     5     5  x  6 


'15' 


85.  Show  that 
4 

15; 

2x2_  4 
3    3x3     9  ; 
5_2x5_10 
21' 


2    2_2x2 
3X5    3x5 

O         O 

3X3"3^3: 

2    5_2x5 
3X 


"3x7 
86.  Show  that 


To  get  f  of  a  fraction,  that 
I   is,    to    multiply  it   by  f ,  we 
|    multiply  the  numerator  by  2, 
and  the  denominator  by  3. 


To  multiply  two  fractions 
together,  we  multiply  the  nu- 
merators together  to  get  a 
new  numerator,  and  the  de- 
nominators together  to  get  a 
new  denominator. 


87.  WhatisJxJ?  Jxf?  ixj?  jxj? 

[Reduce  your  results  to  their  lowest  terms.    See  Ex.  48,  p.  76.] 

88.  What  is^xj?  jxf?  £xj?   |xj?  A*|? 

89.  What  irfxfMx|f  fxA?  fxff  fKf? 

90.  What  is  Iix2f?  1JX3J?  3ix4j?5i 


l> 

2 
9 
3 
2X 
4 
3X 

0 

7 

3 

4 

5 

2 
5 
2 
9 
3 

X 
X 
X 
X 
X 

fi 

7 
2 
3 
3 

12     "I 
"355 
4 

27  5 
9 

2 
4 

X 
X 

4 

2 

8  ' 
8* 

3 

x 

f> 

15' 

91.  A  man  bought  a  piece  of  cloth  for  42^  dol- 
lars, and  was  obliged  to  sell  it  for  J  of  what  it  cost 
him  ;  how  much  did  he  lose  ? 

92.  A  man  bought  a  quantity  of  flour  for  53? 
dollars,  and  sold  it  for  f  of  what  it  cost  him  ;  how 
much  did  he  gain  ? 


90  Fractions.  [§  3. 

93.  If  7  men  can  do  a  piece  of  work  in  4$  days, 
how  long  will  it  take  1  man  to  do  it  ?     How  long 
will  it  take  3  men  to  do  it? 

94.  If  4  men  can  do  a  piece  of  work  in  9^  days, 
how  long  would  it  take  7  men  to  do  it  ? 

95.  There  is  a  pole  standing  so  that  $  of  it  is 
in  the  water,  and  §  as  much  in  the  mud  ;  how  much 
is  in  the  mud  ? 

96.  If  a  man  can  travel  13$  miles  in  3  hours, 
how  many  miles  can  he  travel  in  8  hours  ? 

97.  If  5  horses  can  eat  26  J  loads  of  hay  in  a 
year,  what  will  8  horses  eat  in  the  same  time  ? 

98.  If  4  pipes  can  empty  a  cistern  in  6$  hours, 
how  long  will  it  take  7  pipes  of  the  same  size  to 
empty  it? 


D.  Division  of  Fractions. 

1.  A  boy  having  2  oranges,  wished  to  give  ^  of 
an  orange  apiece  to  his  playmates  ;  how  many  could 
he  give  them  to  ?  If  he  had  given  §  of  an  orange 
apiece,  how  many  could  he  have  given  them  to  ? 

Answer:  If  he  gives  ^  of  an  orange  to  each 
playmate,  he  can  give  to  as  many  playmates  as 
there  are  thirds  in  2,  or  to  6  playmates. 

Had  he  given  §  of  an  orange  to  each,  he  could 
have  given  to  only  half  as  many,  or  to  3  playmates. 

Or,  we  may  find  the  answer  to  the  last  question 
in  this  way  :  In  two  oranges  there  are  6  thirds  ; 
if  he  gives  2  thirds  to  each,  he  can  give  to  as  many 


D.]  Division  of  Fractions.  91 

playmates  as  there  are  twos   in    6,  or  to  3  play- 
mates. 

2.  How  many  times  £  are  there  in  2  ?     How 
many  times  §  are  there  in  2  ? 

3.  A  man  having  3  bushels  of  corn,  distributed 
it  among  some  poor  persons,  giving  |  of  a  bushel 
to  each ;  to  how  many  did  he  give  it  ? 

Answer:  In  3  bushels  there  are  12  fourths 
bushels.  Had  he  given  1  fourth  to  each,  he  could 
have  given  to  12  persons ;  if,  then,  he  gives  3 
fourths  to  each,  he  can  give  to  only  £  as  many,  or 
to  4  persons. 

Or,  since  in  3  bushels  there  are  12  fourths  bush- 
els, he  can  give  3  fourths  to  as  many  persons  as  3 
is  contained  in  12,  or  to  4  persons. 

4.  In  3  are  how  many  times  i  ?  how  many  times 

i? 

5.  If  f  of  a  barrel  of  flour  will  last  a  family 
one  month,  how  long  will  4  barrels  last  the  same 
family  ?     How  long  will  6  barrels  last  ?    How  long 
will  10  barrels  last  ? 

6.  How  many  times  is  §  contained  in  4?  how 
many  times  in  6  ?  how  many  times  in  10  ? 

7.  If  |  of  a  bushel  of  wheat  will  last  a  family 
one  week,  how  many  weeks  will  6|  bushels  last  the 
same  family  ? 

Answer  :  In  6|  bushels  there  are  \7  bushels  ;  at 
the  rate  of  J  in  a  week,  the  whole  would  last  27 
weeks ;  and  at  the  rate  of  j  in  a  week,  the  whole 
would  last  J  of  27  weeks,  or  9  weeks. 

Or,  6 £  bushels  =  V  bushels ;  at  the  rate  of  f  in 


92  Fractions.  [§  3. 

a  week,  ^   will  last  as  many  weeks  as  3  is  con- 
tained in  27,  or  9  weeks. 

8.  How  many  times  is  f  contained  in  6 J  ? 

9.  There  is  a  cistern  having  a  pipe  which  will 
fill  it  in  £  of  an  hour  ;  how  many  times  would  the 
pipe  fill  the  cistern  in  3§  hours?     Ann.  81  times. 

10.  How  many  times  is  §  contained  in  3§? 

11.  How  much  cloth  at  1 J  dollars  (that  is,  f  dol- 
lars) a  yard  can  be  bought  for  4  dollars  ? 

12.  How  many  times  is  l£  or  f  contained  in  4  ? 

13.  How  many  barrels  of  potatoes  at  $lj  a  bar- 
rel can  I  buy  for  $8.1? 

14.  How  many  times  is  1^  contained  in  8i  ? 

15.  If  a  soldier  is  allowed  1^  pounds  (that  is  $  of 
a  pound)  of  meat  in  a  day,  to  how  many  soldiers 
would  6§  pounds  be  allowed? 

16.  How  many  times  is  1:\  contained  in  6§? 

17.  If  If  tons  of  hay  will  keep  a  horse  through 
the  winter,  how  many  horses  will  10  tons  keep  ? 

18.  How  many  times  is  1§  contained  in  10? 

19.  At  2J  dollars  a  box,  how  many  boxes  of 
raisins  can  be  bought  for  10  dollars  ? 

20.  How  many  times  is  2;\  contained  in  10  ? 

21.  At  If  dollars  a  pound,  how  many  pounds  of 
indigo  can  be  bought  for  9|  dollars? 

22.  How  many  times  is  If  contained  in  9|? 

23.  At  If  dollars  a  barrel,  how  many  barrels  of 
raisins  can  be  bought  for  9y  dollars  ? 

24.  How  many  times  is  If  contained  in  9^  ? 

25.  At  I  of    a  dollar   apiece,  how  many  pieces 
of  nankeen  can  be  bought  for  8§  dollars  ? 


D.]  Division  of  Fractions.  93 

26.  How  many  times  is  J  contained  in  8|  ? 

27.  At  f  of  a  dollar  a  pound,  how  many  pounds 
of  tea  can  be  bought  for  7f  dollars  ? 

28.  How  many  times  is  f  contained  in  7J? 

29.  How  many  times  is  83  contained  in  7§? 

30.  How  many  times  is  5^  contained  in  17? 

31.  How  many  times  is  4j  contained  in  9f? 

32.  How  many  times  is  3;J  contained  in 


33.  At    TV  of    a   dollar   a   pound,   how    many 
pounds  of  meat  can  be  bought  for  i  of  a  dollar  ? 

Note.      Change  \  to  tenths. 

34.  How  many  times  is  ^  contained  in  J  ? 

35.  If  a  man  can  do  J  of  a  piece  of  work  in 
one  hour,  how  many  hours  will  it  take  him  to 
do  J  of  the  work? 

Note.     Change  both  fractions  to  twelfths  ;  that 
is,  reduce  them  to  a  common  denominator. 

36.  How  many  times  is  J  contained  in  f  ? 

37.  If  a  pound   of  almonds  cost  \  of  a  dollar, 
how  many  pounds  can  be  bought  for  §  of  a  dollar  ? 

Note.     Reduce  the  fractions  to  a  common  de- 
nominator. 

38.  How  many  times  is  \  contained  in  §  ? 

39.  If  a  piece  of  nankeen  costs  §  of  a  dollar, 
how  many  pieces   can  be   bought  for  4|   dollars, 
that  is,  for  V  dollars  ? 

40.  How  many  times  is  f  contained  in  4|  ? 

41.  If  a  bushel  of  barley  costs   f    of  a  dollar, 
how  many  bushels  can  be  bought  for  f  of  a  dollar  ? 
How  many  for  If  dollars? 


94 


Fractions. 


[§  3. 


contained  in  J  ?   How 


42.  How  many  times  is 
many  times  in  If  ? 

43.  How  many  times  is  ?  contained  in  -;  ? 

44.  How  many  times  is  §  contained  in  £  ? 

45.  Show  that  J  is  contained 
in  1,    3  times  ; 

in  2,    6  times; 

in  3,  9  times  ;  I  To  dl  ^  a  ^hole  n"mb^r  b/ 
in  4,  12  times ;  *  we  multlP1y  the  mimber  b?  3' 
in  5,  15  times. 

46.  Show  that 
in  1,    5  times  ; 
in  2,  10  times ; 
in  3,  15  times ; 
in  4,  20  times. 

47.  Show  that  J 
in  1,  f  times  ;  * 

in  2,  ^  or  5  times ; 
in  3,    *£•  times ; 

3j°  or  10  times. 

48.  Show  that 


in  4, 


is  contained 

To  divide  a  whole  number  by 
i  we  multiply  the  number  by  5. 

is  contained 

To  divide  a  whole  num- 
ber  by  %  we  may  multiply 
the  number  by  5  and  di- 
vide  by  2. 


49.  Show  that 

1^S=  t    [ixl]^  To  divide  a  whole  number 

2-^f=  V°  [f  x  2]  ',  by  a  fraction  we  may  multiply 

3-^J  =  12  [f  x  3]  i  by  the  denominator  and  divide 

5  -=-$  =  ¥  [|x5]J  by    the   numerator,    or,    as    is 

*  |  will  be  contained  only  J  as  many  times  as  £. 


D.]  Division  of  Fractions.  95 

sometimes   said,  we  may   invert  the  divisor  and 
proceed  as  in  multiplication. 

In  the  last  example,  for  instance,  7x5  or  35 
[the  number  of  sevenths  in  5]  is  the  number  of 
times  that  \  is  contained  in  5,  and  ^  will  be  con- 
tained i  as  many  times  or  ^  times. 

50.  Whatis7^-§?  4-f?  6-f?  2(Hf?  21^-J? 
18-fT3T?  16'-=-  ft? 

51.  We  have  already  shown  that 


If  now  we  divide  i  of  2  or  f  by  the  same  divi- 
sors the  results  will  be  only  J  as  large. 
Show,  then,  that 

T|  =  f  x  f  —  A  =  1  0  To  DIVIDE  ONE  FRACTION 
BY  ANOTHER  WE  MAY  IN- 
VERT  THE  DIVISOR  AND  PRO- 
CEED  AS  IN  MULTIPLICATION. 

[Reduce  your  results  to  their  lowest  terms.] 

52.  Whatisf-rV?  §^§?  *^§?  H^l? 

53.  What  is  |-f  A?  if-i7^?  A-5-H? 

54.  Whatis2l-rl|?  2J-5-H?  6i^3i  ?  6-U? 

55.  A  man  bought  6J-  yards  of  cloth  for  |25  ; 
how  much  was  that  a  yard  ? 

56.  A  man  walked  12?-  miles  in  4f  hours  ;  how 
many  miles  an  hour  was  that  ? 

57.  A  man  bought  2£  pounds  of  butter  for  60 
cents  ;  how  much  a  pound  was  that  ? 

58.  At  |  of  a  dollar  a  pound,  how  many  pounds 
of  tea  can  I  buy  for  3  and  f  dollars  ? 


96  Fractions.  [§  3. 

E.  Miscellaneous  Questions. 

1 .  If  A  of  a  ship's  cargo  is  worth  114,000,  what 
is  the  whole  cargo  worth  ? 

2.  A  owiis  ft  of  a  coal  mine,  and  his  share  is 
worth  $ 3,500.    What  is  the  whole  mine  worth  ? 

3.  A  man  willed  J  of  his  property  to  his  wife, 
i  to  a  public  library,  £  to  an  orphan  asylum,  £  to 
his  only  brother,  and  the  remainder  in  equal  shares 
to  his  three  children.     What  part  was  the  share 
of  each  child  ?     If  his  wife's  share  amounted  to 
$  15,220,  what  did  each  of  the  other  shares  amount 
to? 

4.  A  lady  bought  3  pieces  of  cloth ;    the  first 
contained  8J  yards,  the  second  21J  yards,  and  the 
third  15|  yards.     How  many  yards  did  she  buy  in 
all? 

5.  Three  men,   C,  D,  and    E,  bought  a  house 
lot  for  $1488  ;   C  put  in  $248,  D  $744,  and   E 
$496.    What  part  of  the  house  lot  did  each  man 
own  ? 

6.  Four  men,  A,  B,  C,  and  D,  bought  a  mill ; 
A.  contributed  i  the  cost,  B  f ,  C  £,  and  D  $990. 
What  did  the  mill  cost  ? 

7.  A  person  having  sold  £  and  4  of  his  farm 
bad  26 §  acres  left.    How  many  acres  had  he  at 
first  ? 

8.  Which   of   the   following   quantities    is   the 
greatest,  and  which  is  the  least  ?  ^,  ^  -fG  ? 

9.  A  ship  is  worth  $16,000,  and  a  person  who 
owns  YV  °f  it  sells  f  of  his  share  ;  what  share  has 
he  remaining  and  what  is  it  worth? 


E.]  Miscellaneous  Questions.  97 

10.  a.  What  is  f  x  if  ? 

3    16     3x16     48    ,     , 
Note.      |  x  —  =    —     =          (reducing  to  lower 

16    4 

terms)  -op~q 

We  reduced  our  result  to  lowest  terms  by  divid- 
ing both  numerator  and  denominator  first  by  3 
(this  gave  J{j),  and  then  by  4  (this  gave  f).  Now 
we  can  just  as  well  perform  our  divisions  on  the 
expressions  3  x  16  and  4  x  27  as  on  their  equiva- 
lents 48  and  108. 

j  —  =  (dividing  numerator  and  denominator  by 
16  4 

3)  "       (dividins  by  4)=- 


The  following  is  a  short  way  of  expressing  our 
work  :  — 

1  14 

3    16_3xl6_gxl6_g 


_ 

~      ( 


919 
1     4 
316    j|x^    4 


1     9 

In  describing  our  work  we  may  say  :  3  divided 
by  3  is  1,  we  cancel  *  (draw  a  line  through)  the  3 
and  write  the  1  above  it  ;  27  divided  by  3  is  9, 
we  cancel  the  27  and  write  the  9  below  it.  16 

*  This  subject  of  cancellation  is  treated  in  detail  in  Section  X. 
The  little  that  is  said  about  it  here,  however,  will  save  the  pupil 
time  and  labor  in  many  of  the  examples  that  follow. 


98  Fractions.  [§  3. 

divided  by  4  is  4,  we  cancel  the  16  and  write  the 
4  above  it ;  4  divided  by  4  is  1,  we  cancel  the  4 

1x4 

and  write  the  1  below  it.     We  now  have  ^      ^  or 

1  x  j 

a  for  our  answer. 

b.  What   is    A x  i85  ?     [Find   your   answer   by 
cancelling.] 

7       11 
84    121, 


11.  a.  What  is  gg-^    L55-J20- 

77  5      10 


6.  What  is  JSxJf? 

12.  Whatis»x§J?  §§x-H?  JtxJl? 

13.  WhatisH-V?  li-5-A?  A*  A? 

14.  What  is  JJ  x  f  J  x  }£  ? 

15.  What  is  if  XIXA? 

16.  How  many  quarts  of  berries  at  11  cents  a 
quart  will  it  take  to  buy  2jJ  yards  of  cloth  at  16  £ 
cents  a  yard  ?     [2|  x  16*^11  -  3  x  ^  x  ^  =  ?] 

17.  How  many  barrels  of  apples  at  2j  dollars  a 
barrel  will  it  take  to  buy  4j  tons  of  coal  at  5i  dol- 
lars a  ton  ? 

18.  A  man  failing  in  trade  is  able  to  pay  only 
{  of  a  dollar  on  a  dollar ;  how  much  can  he  pay  on 
a  debt  of  832  dollars  ? 

19.  At  $7.86  per  barrel,  what  will  18jj  barrels 
of  flour  cost  ? 

20.  If  a  horse  will  eat  ^  of  &  ton  of  hay  in  a 
month,  how  much  will  80  horses  eat  ? 


E.]  Miscellaneous  Questions.  99 

21.  How  long  will   it  take  a   stage  to  run  4 
miles  if  in  1  hour  it  runs  9J  miles  ?  if  in  1  hour  it 
runs  5  miles?  7J  miles?  6£  miles?  6T7T  miles? 

22.  If  a  family  consume  f  of  a  barrel  of  flour 
in  a  month,  how  long  will  it  take  them  to  use  3 
barrels?  4i  barrels?  7 J  barrels?  3J  barrels?  12f 
barrels  ? 

23.  If  a  man  can  build  15§  rods  of  wall  in  5j 
days,  how  much  can  he  build  in  1  day?  in  7i  days? 
3|  days?  6§  days?  8  days?  j  of  a  day?  §  of  a 
day?  5i  days?  13;\  days?  2g  days? 

24.  If  a  roll  of  carpeting  containing  75  yards 
is  worth  $132,  what  is  |  of  a  yard  worth  ? 

25.  If  a  man  can  perform  a  journey  in   580 
hours,  how  many  days  will  it  take  him  to  perform 
it  if  he  travel  8|  hours  in  a  day  ? 

26.  If  2i  bushels  of  oats  will  keep  a  horse  1 
week,  how  long  will  18$  bushels  keep  him  ? 

27.  If  If,  that  is  |,  of  a  yard  of  cloth  will  make 
a  coat,  how  many  coats  may  be  made  from  a  piece 
containing  6l£  yards? 

28.  If  §  of  a  pound  of  fur  is  sufficient  to  make 
a  hat,  how  many  hats  may  be  made  of  4TV  pounds 
of  fur  ? 

29.  If  lji  yards  of  cloth  is  worth  11  \  dollars, 
what  is  a  yard  worth  ? 

30.  A  merchant  bought  a  piece  of  cloth  con- 
taining 21 1  yards  and  in  exchange  gave  23  J  barrels 
of  flour  ;  how  much  flour  did  one  yard  of  the  cloth 
come  to  ?  How  much  cloth  did  1  barrel  of  the  flour 
come  to  ? 

\ 


SECTION  IV. 
DECIMALS. 

INTRODUCTION. 

Notation  with  simple  illustrative  examples  in 
Addition,  Subtraction,  Multiplication,  and  Di- 
vision. 

This  section  contains  examples  of  fractions  whose  denominators 
are  10,  100,  1000,  etc.  ,  and  points  out  a  simple  method  of  denot- 
ing such  fractions  without  writing  their  denominators. 


(Ill 


The  few  numbers  here 
given  remind  us  that  ev- 
ery time  we  move  a  figure 
one  place  to  the  left  we 
multiply  its  value  by  10  ; 
and  that  every  time  we 
move  a  figure  one  place  to 
the  right  (until  we  reach 
the  units'  place)  we  divide 
its  value  by  10. 

Now  suppose  that,  after 
reaching  the  units'  place, 
we  go  a  step  further  to  the 
right  and  put  the  figure 
6  for  example  in  a  column 
to  the  right  of  the  units'  column  ;  the  principle 


6 

60 
600 
6000 

6000 
600 
60 
6 


ffS3 

6 

06 


6  x  10 

60  x  10 

600  x  10 


6000-10 

600-10 

60-10 


6  4-  10  or 


Introduction.  101 

just  referred  to  leads  us  to  regard  6  in  this  posi- 
tion as  denoting  TV  of  6  or  ft. 

1.  On  the  same  principle,  what  will  the  figure  6 
denote,  if  put  in  the  second  column  to  the  right 
of  the  units'  column? 

2.  What  will  each  of  the  figures  1,  2,  4,  8,  9, 
denote,  if  written  in  the  first  column  to  the  right 
of  the  units'  column  ?  what  if  written  in  the  sec- 
ond column  to  the  right  of  the  units'  column  ? 

3.  What  will  be  denoted  by  1  in  the  first  and  2  in 
the  second  column  to  the  right  of  the  units'  column  ? 

4.  What  name  would  you  give  to  the  first  col- 
umn to  the  right  of  the  units'  column  ? 

5.  What  name  would  you  give  to   the  second 
column  to  the  right  of  the  units'  column  ? 

In  order  to  avoid  writing  the  names  of  the  columns  at  the  top, 
the  following-  device  has  been  adopted :  Take  the  figure  0  for 
example :  When  it  belongs  in  the  first  place  to  the  right  of  the 
units'  place  (that  is  when  it  denotes  tenths)  we  put  a  point  before 
it,  thus  .6  ;  and  when  it  belongs  in  the  second  place  to  the  right 
of  the  units'  place  (that  is  when  it  denotes  hundredths  and  there 
are  no  tenths)  we  put  a  zero  before  it  to  fill  the  vacant  place  ai*d 
a  point  before  the  zero,  thus  .06. 

From  what  precedes  we  know  that  we  may  de- 
note 

ft  by  .1  (sometimes  written  0.1). 

ft  by  .2  (sometimes  written  0.2). 

ft  by  .3  (sometimes  written  0.3). 

}g  by  1.0  or  1 

li  or  1  and  ft  by  1.1 

H  or  1  and  ft  by  1.2 

}%  or  1  and  ft  by  1.3 

6.  Write  in  the  same  way  ft,  ft,  ft,  } $,  H,  }g, 


102  Decimal*.  [§  4. 

7.  What  is  denoted  by  0.7?   0.8?  0.9?  6.7? 
4.6?  12.3?  3.8? 

8.  a.  In  3.2  there  are  32  tenths,  or  3  units  and 
2  tenths  left  over  ;  what   is  the  total  number  of 
tenths  in  each  of  the  following  numbers  ? 

1.7;  1.8;  1.9;  2.7;  4.6;  6.8;  9.2 ;  2  ;  4 ;  3. 
b.  How  many  units  are  there  in  each  of  these 
numbers,  and  how  many  tenths  are  left  over? 

9.  Write  with  a  denominator  and  also  without  a 
denominator  the  sum  of  0.7  and  0.8  (Answers  \$, 
1.5);    the  sum  of  0.6  and  1.6;  the    sum   of    0.3 
and  0.6  ;  the  sum  of  1.2  and  2.6. 

-10.  Write  in  two  ways,  as  in  the  last  example, 
the  answers  to  the  following  : 
.</.  0.4  +  0.8  =  ?         e.  1.2-1.0  =  ? 

b.  0.1  +  0.6  =  ?        /.  0.1  +  0.8  +  0.9  +  0.6  +  1.1  =  ? 

c.  0.9-0.1  =  ?        g.  0.6  +  0.4  +  0.24  1.6  +  0.2  =  ? 

d.  0.3  +  2.0  =  ?        h.  0.9-0.7  +  0.6-0.5  +  0.4  =  ? 

11.  Add  0.2,  0.1,  0.3,  and  0.2. 

We  may,  if  we  wish,  place  these  Q.2 
numbers  in  a  column  as  indicated  on  0.1 
the  right  and  then  add  them  as  we  0.3 
should  add  whole  numbers.  _ 

0.8  Ans. 

12.  Add   0.6,  0.8,  0.4,  and  0.3,  arranging  the 
work  as  in  Example  11.  Ans.  2.1 

13.  What  is  the  sum  of  the  numbers     2.6 
in  the  column  on  the  right  ?  1»8 

The  sum  of  the  tenths  is  21  tenths,     |J** 

or  2  units  and  1  tenth  ;  we  set   down  1_ 

the  1  with  a  point  before  it  and,  adding  15.1  Ans, 


Introduction.  103 

the  2  units  to  the  other  units,  we  get  for  the  entire 
sum  15.1. 

14.  Find  the  sum  of  the  numbers  in  each  of 
the  following  columns : 


a. 

b. 

c. 

d. 

e. 

f. 

9.7 

8.9 

2.2 

6.8 

1.6 

6.4 

6.1 

1.6 

3.3 

4.3 

7.9 

1.0 

8.3 

5.0 

9.9 

7.6 

3.2 

2.7 

1.6 

7.3 

8.0 

3.9 

6.1 

6.2 

0.5 

1.1 

7.6 

1.6 

6.8 

7.9 

0.7 

2.8 

1.9 

0.8 

1.5 

3.2 

Ans.  26.9 

15.  a.  What  is  the  sum  of  the  num-     0.8 
bers  in  the  column  on  the  right  ?  0.7 

0.5 

2.0  Ans. 

6.  What  is  the  total  number  of  tenths  in  this 
sum?  how  many  units  are  there,  and  how  many 
tenths  are  left  over? 

Since  the  answer  amounts  to  exactly  2  units  it  is  not  really  0.8 

necessary  to  write  the  point  and  the  zero ;  it  is  better  to  do  0.7 

so,  however,  in  order  that  our  work  may  not  look  iucom-  Q.5 
plete,  as  if  something-  had  been  accidentally  omitted ;  as  in 

the  column  on  the  right.  ^ 

16.  Write  the  following  numbers  in  a  column, 
tenths   under  tenths  and    units  under  units,  and 
find  their  sum : 

G.4,  8.1,  7.3,  9.8,  6.9,  1.3,  2.1.  Ans.  41.9. 

17.  a.  Write  the  following  numbers  in  a  column 
and  find  their  sum  :     6,  7.1,  1.2,  0.6,  2.3.     Since 
in  6  (the  first  number)   there  are  6  units  and  no 


104  Decimals.  [§  4. 

tenths  left  over  we   may,   if  we  wish, 

place  a  point  after  the  6  and  a  zero  in  y^ 

the  tenths'  place,  thus  6.0  ;  now  writing  ^ 

the  numbers,  tenths  under  tenths  and 

units  under  units,  we  have  the  column  — — : 

on  the  right  AnS'  17'2 

b.  Write  in  the  same  way  the  numbers  7,  2.1, 
1.7,  3,  4.1,  2,  5.1,  1,  6.3,  8,  and  find  their  sum. 

18.  a.  From  »£  take  f  J.     Ans.   ff  or  6.8. 
6.  From  8.6  take  3.8.     Ans.  fg  or  4.8. 

We  may,  if  we  wish,  place  the  smaller  of  these 
two  numbers  beneath  the  larger,  tenths  under 
tenths  and  units  under  units,  and  From  8.6 
then  subtract  as  if  the  numbers  were  take  3.8 
whole  numbers  ;  the  position  of  the  Ans.  4.8 
point  in  the  answer  is  of  course  between  the  units' 
column  and  the  tenths'  column. 

c.  Subtract  in  the  way  just  indicated  7.9  from 
each  of  the  numbers  16.2,  9.7,  11.3,  13.2. 


19. 
20. 

From  16.8 
Take  9.7 

8.4 
6.7 

169.5 
83.7 

98.4 
89.5 

6.2 
1.7 

Ans.  7.1 
a.  From  8 

take  1.6. 

Since  in  8  there  are  8  units  and  no  tenths  left 
over,  we  may,  if  we  wish,  place  a  point  after  the 
8  and  a  zero  in  the  tenths'  place,  thus  8.0.  8.0 

Now  place  the  smaller   number  beneath 
the  larger,  as  indicated  on  the  right.          Ans.   6.4 

b.  From  1  take  0.2 ;  from  16  take  1.9  ;  from  10 
take  2.8  ;  from  80  take  9.6. 

c.  Subtract  17.9  from  each  of  the  numbers  100, 


Introduction.  105 

87.6,  92.1,  23,  18,  49.9,  21,  20.3,  30,  44.6,  1684, 
1728.6. 

21.  Express  both  with  and  without  denomina> 
tors  the  answers  to  the  following  questions : 

a.  8x^  =  ?  (8xA  =  «  or  4.8.) 
5.  9x0.5-?  d.  7x0.6-? 

c.  8x0.4-?  e.  10x0.5-? 

/.  2x6.9-? 

g.  ft-5-3  =  ?  Glw*.  A  or  0.3.) 
A.  0.8-7-2  =  ?  j.  1.8-2-? 

i.  0.6-3-?  L  3.6^3  =  ? 

22.  Multiply  each  of  these  numbers  by  10  : 
0.1  (Ans.  \%  or  1)  ;  0.3  (Ans.  ?#  or  3)  ;  0.7 ; 

0.9  ;  1.1  (Ans.  W  or  11)  ;  1.5  ;  1.9  ;  3.8  ;  16.1 ; 
18.1 ;  96.4  ;  184.6  ;  192.9. 

23.  What  is  the  quotient  of  A^A?    (A^A  = 
AxJ^  =  4*)  TV*-A?   O1^-    3)    0.4^-0.2?  0.2^- 
0.2?  7.5-^0.3?  0.9-f  0.3?  2.7^-0.3?  98.4^32.8? 

24.  What  is  the  quotient  of  8^0.2?  (8-r0.2  = 
8^-^-8x^0^=40)  6^0.3?  (^7^.20)  16^0.8? 
27-^0.9?  144^-1.2?   2^-0.2?  4^0.2?  98.1^0.9? 
5.7^-1.9? 

25.  Divide  each  of  the  following  numbers  by 
10,  expressing  the  answers  both  with  and  without 
denominators : 

1  (1^-10  =  ^  or  0.1)  ;  64  (64-=-10  =  ?J  =  6.4)  ;  3  ; 
6  ;  123 ;  172  ;  968 ;  1728. 

*  A  g-ood  knowledge  of  common  fractions  is  here  indispensa- 
ble ;  the  student  should  test  the  accuracy  of  each  step  for  him- 
self, illustrating,  if  necessary,  by  simple  examples  of  his  own, 
and  he  should  not  be  allowed  to  state  a  rule  except  as  the  result 
of  his  own  fresh  experience. 


106  Decimals.  [§  4. 

Now  advancing  one  step  further  to  the  right, we 
may  denote 

Tfo  by  .01  or  0.01 
Tfa  by  .02  or  0.02 
Tfo  by  .03  or  0.03 
TV&  by  .11  or  0.11 
TV*  by  .12  or  0.12 
m  by  1.00  or  1 
Hi  by  1.01 
m  by  1.02 
HJ  by  1.11 
m  by  1.12 
W%  by  2.00  or  2 

26.  Write  in  the  way  indicated  above  yj^,  rg^, 

T$U>  T<b>  lV%  T^»  !*&»  H$»  ?§$• 

27.  What  is   denoted  by  0.09?    0.23?    0.98? 
1.82?  3.21?  0.90?  0.70. 

28.  In  1\J  there  are  TW  ;  we  may  say,  then,  that 
0.1  is  equal  to  0.10.     How  many  hundredths  are 
there  in  each  of  the  following  numbers  ?   Express 
the  answers  without  denominators,  0.7  (Ans.  0.70)  ; 
0.9;  0.2;  0.6;  0.8. 

29.  What  is  the  sum  of  0.7  and  0.08?   (0.7  is 
equal  to  0.70  and  this  added  to  0.08  gives  0.78) 
What  is  the  sum  of  0.6  and  0.09  ?  (Ans.  0.69) 
of  0.5  and  0.13?  of  0.4  and  0.09?  of  0.8  and  0.62? 
of  1.2  and  0.32  ? 

30.  a.  In  0.78  there  are  7  tenths  and  8  hun- 
dredths.    How  many  tenths  are  there  in  each  of 
the  following  numbers,  and  how  many  hundredths 
are  left  over  ?  0.69 ;  0.60  ;  0.53  ;  0.48  ;  0.87. 


Introduction.  107 

b.  What  is  the  total  number  of  hundredths  in 
each  of  the  preceding  numbers  ? 

31.  a.  How  many  units  are  there  in  each  of  the 
following  numbers,  and  how  many  hundredths  are 
left  over  ?  1.86   (Ans.  1  unit  and  86  hundredths 
over);  2.03;  1.68;  17.64;  9.10;  38.42;  6;  1;  4. 

b.  What  is  the  total  number  of  hundredths  in 
each  of  the  preceding  numbers  ?  [In  the  first 
(1.86)  there  are  186  hundredths.] 

32.  Express  with  a  denominator  and  also  with- 
out a  denominator  the  answers  to  the  following 
questions :  — 

a.  0.64  +  0.29  =  ?  Answers  fl&;  0.93 

b.  0.67  +  0.11  =  ?        /.  0.5  +  0.91  =  ? 

c.  0.32  +  0.16  =  ?        g.  0.4  +  0.2  +  0.18=? 

d.  0.30  +  0.12=?       h.  0.6  +  0.66  +  0.53  =  ? 

e.  0.6  +  0.17=?  i.  4  +  0.4  +  0.04  =  ? 

33.  Add  the  numbers  in  each  of  the  following 


columns  : 

a. 

6. 

c. 

d. 

e. 

/• 

0.12 

6.82 

0.01 

1.61 

8.4 

1.11 

0.68 

1.61 

1.01 

.80 

6.84 

1.23 

0.73 

0.05 

0.10 

.94 

7.32 

4.56 

0.96         .70      8.76       1.79       9.99      7.89 

34.  From       3.68     1.83     6.07     1.08     0.99 
take         1.72       .76     1.02       .18       .02 

Answer :     1.96 

35.  What  is  the  product  of  8  x  Tfo  ?  (Ans.  ffo 
or  0.48)   7  x  ^  ?  5  x  0.06  ?  9  x  0.16  ?  4  x  0.02  ? 
25x0.25? 


108  Decimals.  [§  4. 

36.  Multiply  each  of  the  following  numbers  by 
100:  0.06    (Ans.   0. 06  =  T-|fo,  therefore  0.06x100 
=  TfoxlOO  =  6);  0.09;  0.16;   0.89;  1.16;  9.84; 
11.84 ;  69.42. 

37.  What  is  the  product  of  A  x  ^  ?    (Ans.  ffo 
or  0.32)  Ax  A?  0.4x0.9?   1.6x1.6?  2.5x2.5? 
0.9x8.1? 

38.  a.  Give  three  new  examples,  like  those  of 
No.  37,  of  tenths  multiplied  by  tenths. 

b.  When  we  multiply  tenths  by  tenths,  what  is 
the  denominator  of  the  product  ? 

c.  When  the   product   is   expressed   without  a 
denominator,  how  many  figures  are  there  to  the 
right  of  the  point  ? 

39.  Multiply  0.6  by  0.9 ;  0.8  by  1.6  ;  0.7  by 
1.7  ;  2.3  by  7.6  ;  1.2  by  0.4. 

40.  Multiply  0.8  by  0.6  ;  0.9  by  1.7  ;  0.4  by 
0.3  ;  1.4  by  1.6  ;  8:5  by  6.2. 

41.  What  is  the  quotient  of  Tf^2?  (Ans.  Tfo 
or  0.04)  TVfc-s-8?  y^-7? 

42.  What   is  the  quotient  of  0.06-^3?    (Ans. 
0.02)  0.08-J-4?  1.44 -M2? 

43.  What  is  the  quotient  of  0.8^10?   (0.8  is 
the  same  as  0.80  and  0.80^-10-0.08)   0.9^-10? 
1.2-5-10?  14.4^10?  172.8-rlO? 

44.  What  is  the  quotient  of  2-flOO?  (2  is  the 
same  at  2.00  or  f gg,  and  this  divided  by  100  is 
T§o  or  0.02)  12-^100?  168-rlOO?  1123^-100? 

45.  What  is  the  quotient  of  T§^  -r  A  ?   (T§TF  "^ 
T%-TSox-V°-::=A  or  0.3)      What  is  the  quotient 


Introduction.  109 

of  0.08-0.4?  (Ans.  &  or  0.2)  0.16-K8?  0.18- 
0.9?  0.27-^0.3?  0.24^0.8?  0.09^0.3?  0.22^0.2? 
0.27-0.9?  0.12*0.6? 

46.  a.  Give  three  new  examples,  like  those  of 
No.  45,  of  hundredths  divided  by  tenths. 

6.  When  we  divide  hundredths  by  tenths,  what 
is  the  denominator  of  the  quotient  ? 

c.  When  the  quotient  is  expressed  without  a 
denominator,  how  many  figures  are  there  to  the 
right  of  the  point  ? 

47.  Divide  0.33  by  1.1  (Ans.    0.3)  ;  0.24  by 
1.2  (Ans.   0.2)  ;  0.48  by  1.2 ;  0.36  by  1.2  ;  0.28 
by  1.4  ;  0.48  by  2.4  ;  0.32  by  1.6  ;  0.22  by  1.1. 

48.  What  is  the  quotient  of  T^-y^?  (Ans.  4) 
TVfe-M*?  0.21-f0.07?  0.84-0.21?  16.38-f8.19? 

49.  What  is  the  quotient  of    8  -f  T§D  ?    (^Us. 
400)   6^0.03?  (Ans.  200)  16^-0.08?  27^-0.09? 
18-0.03  ? 

Advancing  another  step  to  the  right,  we  may 
denote 

by  -001  or  0.001 
by  .002  or  0.002 
by  .011  or  0.011 
by  .060  or  0.060 
by  .684  or  0.684 
HU  by  1.111 

by  2.000  or  2 
by  16.213 

50.  Write  without  the  denominators 


110  Decimals.  [§  4. 

51.  Write  without  the  denominators  Ti8^»  TiSiT> 

iMtfi  TfffflH  T$$ff5  T&<hy     T&8ff« 

52.  Write  without  the  denominators 


53.  In  8.467  there  are  8  units,  4  tenths,  6  hun- 
dredths,  and  7  thousandths.     In  like  manner,  give 
the  value  of  each  figure  in  the  numbers  6.382, 
0.379,  1.843,  98.432,  168.017,  3.169,  8.006,  1.111, 
666.666. 

54.  Add  the  numbers  in  each  of  the  following 
columns  :  — 

a.            b.  c.           d.           e. 

0.684  0.968  1.123  9.842  8.213 

0.769  0.432  2.345  8.731  9.324 

0.943  0.179  3.456  7.620  0.435 

0.210  6.842  4.678  6.519  1.546 

Ans.  2.606 

55.  From          0.842     6.842     3.986     9.610 
take  0.675     3.986     1.997     8.432 

Ans.  0.167 

56.  Place  the  following  numbers  in  a  column 
and  find  their  sum  :  6,  0.6,  0.06,  0.006. 

Show  that  6   may  be  written  6.000  ; 

6  that  0.6  may  be  written  6.000 

0.6         0.600  ;  and  that  0.06  may  0.600 

0.06       be  written  0.060.     The  0.060 

0.006     column  of  numbers  may  0.006 

Ans.  6.666     then  be  written  as  indi-  Ans.  6.666 
cated  on  the  right. 


Introduction.  Ill 

57.  Place  the  following  numbers  in  a  column, 
as  in  the  last  part  of  Example  56,  and  find  their 
sum  :  6,  0.016,  0.76,  7.16,  0.171,  9.842 

58.  Add  7.942,  6.77,  4.3,  1.842,  0.009,  0.07, 
and  1.111 

59.  What  is  the  product  of  AXTJTF?     (Ans. 
rfcta,  or  0.032)  ^  x  T fo  ?   0.4  x  0.09  ?   0.3  x  0.84  ? 
0.7x0.68?  2.4x6.84? 

60.  a.  Give  three  new  examples,  like  those  of 
No.  59,  of  tenths  multiplied  by  hundredths. 

b.  When   we    multiply   tenths    by   hundredths, 
what  is  the  denominator  of  the  product  ? 

c.  When    we    express   the    product   without  a 
denominator,  how  many  figures  will  there  be  to  the 
right  of  the  point  ? 

61.  Multiply  0.6  by  0.17  (An*.    0.102);    0.9 
by  0.84    (Am.    0.756);    1.9    by  8.64;    0.8    by 
0.04  ;  1.8  by  0.14  ;  0.9  by  1.01 ;  0.6  by  0.06  ;  0.8 
by  0.08  ;  9.86  by  8.4. 

62.  What  is  the  product  of  0.009  x  10  ?     (Ans. 
0.09)    0.09x10?    (Ans.   0.9)    0.9x10?   9x10? 
90  x  10  ? 

63.  What  is  the  product  of  0.864  x  10  ?     (Ans. 
8.64)  8.64x10?  86.4x10?  864x10? 

64.  By  what  must  we  multiply  0.006  in  order 
to  get  0.06  ?  (Ans.   10)  0.06  to  get  0.6  ?   0.6  to 
get  6  ?  6  to  get  60  ? 

65.  When  we  multiply  a  number  by  10,  how 
many  places,  with  reference  to  the  point,  do  we 
move  the  number  to  the  left  ?     Illustrate  by  3  new 
examples. 


112  Decimals.  [§  4. 

66.  Multiply  6.642  by  10.    Ans.  66.42.      We 
may  say  that  the  answer  is  found  by  moving  the 
number  one  place  to  the  left,  with  reference  to  the 
point,  or,  what  amounts  to  the  same  thing,  by  mov- 
ing  the   point  one  place  to  the  right.     Multiply 
0.008  by  10;  0.073  by  10;  23.6  by  10;  86.4  by 
10;  1.618  by  10. 

67.  Divide  682  by  10  ;  68.2  by  10 ;  6.82  by  10. 

68.  By  what  must  we  divide  60  to  get  6  ?  6  to 
get  0.6  ?  0.6  to  get  0.06?  0.06  to  get  0.006? 

69.  When  we  divide  a  number  by  10,  how  many 
places,  with  reference  to  the  point,  do  we  move  the 
number  to  the  right?    Illustrate  by  3  new  examples. 

70.  Divide  6.4  by  10 :  6.4  -r  10  =  0.64    Ans. 
We  may  say  that  the  answer  is  found  by  moving 
the  number  one  place  to  the  right,  with  reference 
to  the  point,  or,  what  amounts  to  the  same  thing, 
by  moving  the  point  one  place  to  the  left.     Divide 
0.4  by  10  ;  0.04  by  10;  2.16  by  10 ;  12.31  by  10  ; 
1231  by  10. 

71.  What  is  the  product  of  0.009  x  100  ?  (Ans. 
0.9)  0.09x100?  0.9x100?  9x100?  0.682x100? 
6.82x100?  68.2x100?  682x100?  6820x100. 

72.  By  what  must  we   multiply  0.006  to  get 
0.6  ?  ( Ans.  100)  0.06  to  get  6  ?  6  to  get  600  ? 

73.  When  we  multiply  a  number  by  100,  how 
many  places  to  the  right  do  we  move  the  point  ? 

74.  Multiply   6.666   by   100;   89.61   by  100; 
0.723  by  100  ;  0.07  by  100  ;  0.009  by  100  ;  8.006 
by  100  ;  8.06  by  100. 

75.  Divide  900  by  100  ;  90  by  100  ;  9  by  100  ; 
0.9  by  100. 


Introduction.  113 

76.  By  what  must  we  divide  600  to  get  6  ?  60 
to  get  0.6  ?  6  to  get  0.06  ?  0.6  to  get  0.006  ? 

77.  When   we  divide  a  number  by  100,  how 
many  places  to  the  left  do  we  move  the  point  ? 
Give  a  new  example. 

78.  Divide  6849  by  100  ;  734.6  by  100 ;  8  by 
100  ;  0.8  by  100  ;  68.7  by  100. 

Advancing  another  step  to  the  right,  we  may  de- 
note 

T**™  by  0.0001 
Tiftftnr  by  0.0088 
by  0.0161 
by  0.8432 
by  1.6842 

Advancing  another  step  to  the  right,  we  may  de- 
note in  the  same  way 

by  0.00001 
by  0.00088 
by  0.00161 
by  0.08432 
by  0.16842 

79.  Add    6,    0.6,    0.06,    0.006,    and    0.0006 
Answer:  6.6666 

80.  Add  the  numbers  in  each  of  the  following 
columns  : 

4.0000  0.0900  8.4321  1.3264 

0.6000  1.8402  7.6176  0.4421 

0.1830  0.0673  0.0094  0.6086 

0.6842  0.1162  0.0009  0.0798 

AM.  5.4672 


114 


Decimals. 


[§4. 


81.  Add  0.09,  7.6123,  0.0092,  and  0.0001 

82.  Add  8.998,  0.6132,  4,  and  2.16 

Below  are  given  the  names  of  each  of  the  first 
nine  places  to  the  left,  and  to  the  right  of  the 
units'  place. 


6666666666.G66666666 

86.9 

87.03 
8.33 

243.006     * 
2.684 
6.0007 
8.6842 
0.00004 
1.68426 
1.000006 
4.321007 
0.0000009 
0.16824161 
8.001018423 


TO   BE   READ 

86,  and  9  tenths. 

87,  and  3  hundredths. 
8,  and  33  hundredths. 

243,  and  6  thousandths. 
2,  and  684  thousandths. 
6,  and  7  ten-thousandths. 
8,  and  6842  ten-thousandths. 
4  hundred-thousandths. 
1,  and  68426  hundred-thousandths. 
1,  and  6  millionths. 
4,  and  321007  millionths. 
9  ten-million ths. 
16824161  hundred-millionths. 
8,  and  1018423  billionths. 


83.  Read  each  of  the  following  numbers  : 
a.  1.69  {Ans.  one,  and  sixty-nine  hundredths) ; 
b.  0.721  (Ans.  seven  hundred  twenty-one  thou- 
sandths) ;  c.  8.6346  (Ans.  eight,  and  six  thou- 
sand three  hundred  forty-six  ten-thousandths) ; 
d.  168.04 ;  e.  1.006004  ;  /.  21.69267  ;  g. 
101.000101 ;  h.  0.841682713 ;  i.  0.000069247  " 

*  Read,  Two  hundred  forty-three,  not  two  hundred  and  forty- 
three  ;  in  a  case  of  this  sort  it  is  better  not  to  use  the  word  and 
except  between  whole  numbers  and  decimals. 


Introduction.  115 

84.  Express  by  figures,  without  using  denomi- 
nators, the  numbers :  8,  and  4  tenths  ( Ans.  8.4)  ; 
98,  and  44  hundredths  (Ans.  98.44)  ;    6,  and  9 
thousandths    (Ans.    6.009)  ;    8    ten-thousandths ; 
1,  and   82    hundred-thousandths ;    864   millionths 
(Ans.    0.000864)  ;     2,    and    116   ten-miUionths ; 
6013     hundred-millionths  ;     8,     and     684721986 
billionths ;  2,  and  13  thousandths ;  849  ten-thou- 
sandths ;    23   thousandths  ;     1564   millionths;    9 
hundred-thousandths  ;  231  thousandths. 

85.  Read  the  numbers  : 
7,211,136,298.033  (Ans.  1  billion  211  million 

136  thousand  298,  and  33  thousandths)  ;  211,136,- 
298.017;  11136298.1698;  1136298.11116;  136- 
298.184321 ;  36298.2432691 ;  6298.6  ;  298.10237- 
612;  98.0000006  (Ans.  98,  and  6  ten-millionths)  ; 
8.100068421 ;  1.000006. 

86.  Express  by  figures,  without  using  denomina- 
tors, the  numbers : 

263,  and  198  thousandths  (Ans.  263.198); 
1,  and  6241  ten-thousandths  ;  2301,  and  168  hun- 
dred-thousandths ;  145932  millionths ;  24,  and 
333  ten-millionths:  2846972  hundred-millionths; 
2846972  billionths. 

87.  Express  by  figures,  without  using  denomina- 
tors, the  numbers : 

3,  and  28  billionths  ;  13  ten-millionths  ;  564 
ten-thousandths  ;  1  thousand,  and  1  thousandth  ; 
114,  and  114  thousandths  ;  6,  and  6  hundred-mil- 
lionths; 743  hundred-thousandths;  1722,  and  1722 
millionths ;  69,  and  84  hundredths. 


116  Decimals. 

88.  60000. 
6000. 

600.  In  the  adjacent  column  each 

fiO 

number   is   how    many   times 

Q*/>          the  number  just  below  it  and 

o!o6        *s  wnat   Part  °f  the  number 

0.006      just  above  it? 

0.0006 

0.00006 

NOTE.  We  know  that  the  value  of  a  figure  is  multiplied  by 
TEN  by  moving  it  one  place  to  the  left  (with  reference  to  the 
point) ;  and  that  this  value  is  consequently  divided  by  TEN  by 
moving  the  figure  one  place  to  the  right ;  it  is  on  this  account  that 
the  system  of  numbers  we  have  used  is  called  the  ten  or  decimal  * 
system  of  numbers ;  the  point  or  period  placed  between  the  unit's 
place  and  the  tenth's  place  is  called  the  decimal  point,  and  the 
fraction  to  the  right  of  the  point  is  called  a  decimal  fraction. 

*  The  word  decimal  is  from  the  Latin  decem,  which  means  ten. 


SECTION  V. 
MULTIPLICATION   OF  DECIMALS. 

A.  Multiplication  of  a  Decimal  by  a  Whole 
Number.  Examples  and  Problems  with  He- 
marks  and  Explanations. 

1.  a.  Multiply  9.6  by  4. 

Applying  the  method  used  in  the  multiplication 
of  whole  numbers  we  say  4  QQ  Multiplicand, 
times  o  tenths  are  24  tenths,  4  Multiplier. 

or  2.4 ;  write  the  .4  vertically     T^~ 

'  J,     38.4  Product, 

under  the  .0  and  save  the  2 

[units].  4  times  9  are  36 ;  add  the  2  that  were 
saved  and  write  the  result  38  to  the  left  of  the  point. 
I).  Show  that  the  product  of  9.6  and  4  is  38.4, 
by  first  reducing  the  9.6  to  a  common  fraction  and 
then  proceeding  as  in  the  multiplication  of  a  frac- 
tion by  a  whole  number. 

2.  Multiply  1.842  by  4,  in  each  of  the  two  ways 
just  indicated.  Ans.  7.368 

3.  Multiply  1.842  by  6,  in  two  ways. 

Ans.  11.052 

4.  Multiply  0.1842  by  9,  in  two  ways. 

Ans.  1.6578 

5.  a.  How  many  decimal  places  are  there  in  the 


118  Decimals.  [§  5. 

multiplicand,  and  how  many  in  the  product  in  each 
of  the  last  four  examples  ? 

b.  Find  the  product  of  1.842  and  12. 

6.  a.  Find  the  answer  to  each  of  the  following 
examples,  and  see  if  you  have  the  same  number  of 
decimal  places  in  the  product  as  in  the  multipli- 
cand. 

b.  Take  any  number  you  please  containing  one 
decimal  place  and  multiply  it  by  7. 

c.  Take  any  number  you  please  containing  two 
decimal  places  and  multiply  it  by  8. 

d.  Take  any  number  you  please  containing  three 
decimal  places  and  multiply  it  by  11. 

e.  Take  any  number  you  please  containing  four 
decimal  places  and  multiply  it  by  6. 

NOTE.  From  what  precedes  we  see  that  when  the  multiplier  is 
a  whole  number  we  may  multiply  exactly  as  in  whole  numbers, 
pointing  off  the  same  number  of  decimal  places  in  the  product  as 
there  are  in  the  multiplicand. 

7.  Multiply  184.2  by  144. 

8.  Multiply  1.842  by  144. 

9.  Multiply  8.42316  by  32. 

10.  Multiply  16.82  by  39. 

11.  Multiply  0.693  by  842. 

12.  Multiply  0.0396  by  97. 

13.  Multiply  0.0064  by  4. 

14.  How  many  gills  are  there  in  0.35  pints? 

Ans.  1.4  gi. 

15.  How  many  pints  are  there  in  0.125  quarts? 
(Ans.  0.25  pts.)  how  many  gills?  (Ans.  1  gi.) 

16.  How  many  quarts  are  there  in  0.0125  gal- 
lons ?  how  many  pints  ?  how  many  gills  ? 


A.]  Multiplication.  119 

17.  How  many  gills  are  there  in  0.6875  gallons  ? 

18.  Show  that  0.9375  gallons  =  3  qts.  1  pt.  2  gi. 
First:  Reducing  to   quarts,  we   get  0.9375  gal- 
lons =  4  x  0.9375  qts.  =  3.75  qts.      Second:  Redu- 
cing the  .75  qts.  to  pints,  we  get  .75  qts.  =  2x  .75 
pts.  =  1.5  pts.     Third  :    Reducing  the   .5  pts.  to 
gills,  we  get  .5  pts.  =  4x.5  gi.  =  2  gi.     .-.  0.9375 
gallons  =  3  qts.  1  pt.  2  gi.     This  is  called  reducing 
the  decimal  part  of  a  gallon  to  quarts,  pints,  and 
gills. 

The  work  may  be  more  briefly  arranged  as  fol- 
lows: 

0.9375  gallons. 
4 

3.7500  qts. 

_2 

1.50  pts. 
4 


19.  Reduce  0.175  gallons   to  quarts,  pints,  and 
gills.     Ans.  0  qts,  1  pt.  1.6  gi. 

20.  Reduce  0.435  gallons  to  quarts,  pints,  and 
gills. 

21.  Reduce  0.2175  gallons  to  quarts,  pints,  and 
gills. 

22.  What  is  the  cost  of  2  quarts  of  alcohol  at 
11  cents  a  gill  ? 

23.  What  is  the  cost  of  24  gallons  of  vinegar 
at  7  cents  a  quart  ? 

24.  What  is  the  cost  of  8  two-gallon  cans  of 
milk  at  6  cents  a  quart  ? 


120  Decimals.  [§  5. 

25.  What  is  the  cost  of  3  gallons  of  ice-cream 
at  15  cents  a  quart? 

26.  What  is  the  cost  of  3  gallons  of  astral  oil 
at  10.0625  a  quart? 

27.  How  many  pecks  are  there  in  0.35  bushels? 
how  many  quarts  ?  how  many  pints  ? 

28.  How  many  pints  are  there  in  1.45  bushels  ? 

29.  Reduce  0.109375  bushels  to  pecks,  quarts, 
and  pints. 

30.  Reduce  0.809375  bushels  to  pecks,  quarts, 
and  pints. 

31.  Reduce  0.984375  bushels  to  pecks,  quarts, 
and  pints. 

32.  How  many  pints  are  there  in  0.02  bushels  ? 

33.  At  12  cents  a  quart,  what  will  3  pecks  of 
berries  cost  ? 

34.  At  $0.145  a  quart,  what  will  a  bushel  of 
grass  seed  cost  ? 

35.  At  $0.1875  a  peck,  what  will  2  bushels  of 
potatoes  cost  ? 

36.  At  $0.285  a  peck,  what  will  3  bushels  of 
apples  cost  ? 

37.  How  many  pounds  are  there  in  0.001  tons? 
how  many  ounces  ? 

38.  How  many  ounces  are  there  in  0.0025  tons? 

39.  Reduce  0.00125  tons  to  pounds  and  ounces. 

40.  Reduce  0.0006  tons  to  pounds  and  ounces. 

41.  Reduce  0.21512  tons  to  pounds  and  ounces. 

42.  At  $0.345  a  pound,  what  will  14  pounds  of 
coffee  cost? 

43.  At  $0.00275  a  pound,  what  will  a  ton  of 
range  coal  cost  ? 


A.]  Multiplication.  121 

44.  At  $ 0.006  a  pound,  what  will  a  ton  of  Amer- 
ican Cannel  coal  cost? 

45.  At  $0.011  a  pound,  what  will  a  ton  of  hay 
cost? 

46.  At  10.115  a  pound,  what  will  •    tons  of  lead 
cost? 

47.  How  many  feet  are  there  in  0.001  miles  ? 
how  many  inches? 

48.  How  many  rods  are  there  in  0.12525  miles  ? 
how  many  yards  ?   how  many   feet  ?   how   many 
inches  ? 

49.  Reduce  0.002  miles  to  rods,  yards,  feet,  and 
inches. 

50.  Reduce   0.5135   miles   to  rods,  yards,  and 
feet. 

5 1 .  What  would  it  cost  to  build  38  rods  of  fence 
at  10.305  a  foot? 

52.  What  would  2  miles  of  telegraph  wire  cost 
at  $0.003  a  foot? 

53.  What  would  108  fathoms  of  rope  cost  at 
$0.0125  a  foot  ?     [A  fathom  =  6  f t.] 

54.  What  would  be  the  cost  of  enough  steel  rails 
for  a  mile  of  a  single  track  railroad  at  $0.055  a 
foot  ? 

55.  If  sound  travels  a  foot  in  0.0008  seconds, 
how  long  will  it  take  it  to  travel  10  miles? 

560  Reduce  0.615  miles  to  rods,  yards,  feet,  and 
inches. 

57.  Reduce  0.115  miles  to  rods,  yards,  feet,  and 
inches. 

58.  Reduce  0.115125  miles  to  rods,  yards,  feet, 
and  inches. 


122  Decimals.  [§  5. 

59.  Reduce  0.002125  miles  to  rods,  yards,  feet, 
and  inches. 

60.  Reduce  0.5  rods  to  hands.    [A  hand  =  4  in.] 

61.  Reduce  0.0025  miles  to  hands. 

62.  How  many  seconds  are  there  in  0.01  hours  ? 

63.  How  many  minutes  are  there  in  0.05  days? 

64.  Reduce  0.01  years  to  days,  hours,  minutes, 
and  seconds. 

65.  Reduce  0.0005  years  to  days,  hours,  min- 
utes, and  seconds. 

66.  Reduce  0.0004  years  to  days,  hours,  min- 
utes, and  seconds. 

B.  Multiplication  of  a  Decimal  by  a  Decimal. 
Examples  and  Problems,  with  Remarks  and 
Explanations. 

1.  Multiply  18.42  by  1.2 

Here  the  multiplier  1.2  is  the  same  as  \$ ;  there- 
fore 18.42  x  1.2  - 18.42  x  ^  =  2-^Q4  =  22.104  Ans. 

2.  Multiply  6.82  by  0.2  Ans..  1.364 

3.  Multiply  36.7  by  0.6  Ans.  22.02 

4.  Multiply  0.067  by  0.4  Ans.  0.0268 

5.  a.  In  Example  1   we  first  multiplied  by  12 
and  then  divided  by  10 ;  now  the  effect  of  dividing 
by  10  is  merely  to  move  the  point  one  place  to  the 
left.     We  may   say,  therefore,   that   in    order  to 
multiply  18.42  by  1.2,  we  may  first  multiply  by  12 
and  then  move  the  point  one  place  to  the  left  in 
the  product.     We   may   arrange  our  work  as  fol- 
lows : 


B.]  Multiplication.  123 

18.42  18.42  Multiplicand. 

12  1.2  Multiplier. 

3684       or  more  briefly      3684 
1842  1842 

221.04  22.104  Product. 

22.104  Ans. 

b.  Multiply  1.732  by  1.4 

1.732  c.  Multiply  267.3  by  1.6 

1.4  Ans.  427.68 

6928 
1732 


2.4248  Ans. 

6.  Multiply  184.2  by  0.4.     Ans.  73.68 

7.  Multiply  1.842  by  .6 

8.  Multiply  18.42  by  0.8 

9.  Multiply  0.695  by  13.2 

10.  Multiply  0.00763  by  3.8 

11.  Multiply  7.614  by  38.2 

12.  When   there  is  one  decimal   place    in  the 
multiplier  (as  in  each  of  the  last  six  examples),  how 
many  more  decimal  places  are  there  in  the  product 
than  in  the  multiplicand,  and  why?    How  many 
decimal  places  will  there  be  in  the  product  in  each 
of  the  following  examples,  and  why  ? 

a.  18.42x12.3  c.  2.135x1.3 

b.  69.1x1.1  c?.  0.1684x1.4 

13.  Find  the  product  in  each  of  the  cases  just 
given. 

14.  a.  Multiply  18.42  by  0.06 


124  Decimals.  [§  5. 

Here  the  multiplier  0.06   is  the  same  as  jg^  ; 
therefore   18.42  x  0.06  =  18.42  x  ^  =  Uftp  = 
1.1052    Ans. 

b.  Multiply  1.842  by  0.12  Ans.  0.22104 

15.  Multiply  1.842  by  0.08  Ans.  \A~CM 

16.  Multiply  18.432  by  0.09  Ans.  1.65888 

17.  Turning  back  to  Example  14  a.  we  see  that 
the  last  step  in  the  process  consists  in  dividing  by 
100,  the  effect  of  which  is  merely  to  move  the 
point  two  places  to  the  left.     We  may  say,  then, 
that  in  order  to  multiply  18.42  by  0.06  we  may 
first  multiply  by  6  and  then  move  the  point  two 
places  to  the  left  in  the  product.     We  may  ar- 
range the  work  briefly,  as  follows  : 

18.42  Multiplicand. 
0.06  Multiplier. 

1.1052  Product. 
a.  Multiply  18.42  by  1.23 

18.42  b.  Multiply  0.2268  by  0.16 

1.23  Ans.    3.06288 

5526 
3684 
1842 


22.6566    Ans. 

18.  Multiply  16.8  by  0.03  Ans.  0.504 

19.  Multiply  144  by  0.12 

20.  Multiply  111.234  by  0.09 

21.  When  there  are  two  decimal  places  in  the 
multiplier  (as  in  each  of  the  last  eight  examples), 
how  many  more  decimal  places  are  there  in  the 


B.]  Multiplication.  125 

product  than  in  the  multiplicand,  and  why  ?  How 
many  decimal  places  will  there  be  in  the  product 
in  each  of  the  following  cases,  and  why  ? 

a.  1.842x1.23  c.  69.1x0.11 

b.  1.842  x  0.32  d.  0.6842  x  1.21 

22.  Find  the  product  in  each  of  the  cases  just 
given. 

23.  Multiply  18.42  by  0.006         Ans.  0.11052 

24.  Multiply  2.634  by  0.004        Ans.  0.010536 

25.  Multiply  111.32  by  0.015 

26.  Why  is  it  that  when  there  are  three  decimal 
places  in   the   multiplier  the  number  of   decimal 
places  in  the  product  is   three  more  than  in  the 
multiplicand  ? 

27.  How  many  decimal  places  will  there  be  in 
the   product  in  each  of  the  following  cases,  and 
why  ? 

a.  184.2x0.012  c.  1.234x1.112 

b.  6.91x0.006  d.  162x0.016 

28.  Find  the  product  in  each  of  the  cases  just 
given.     Below  is  given  the  work  for  the  first  case : 

a.  184.2 
0.012 

3684 
1842 


2.2104  Ans. 

29.  Multiply  1164.1  by  0.0006      Ans.  0.69846 

30.  Multiply  1762  by  0.0012  Ans.  2.1144 

31.  Multiply  236.8  by  0.1112      Ans.  26.33216 

32.  Why  is  it  that  when  there  are  four  decimal 


126  Decimals.  [§  5, 

places  in  the  multiplier  the  number  of  decimal 
places  in  the  product  is  four  more  than  in  the 
multiplicand  ? 

33.  How  many  decimal  places  are  there  in  the 
product  in  each  of  the  following  cases,  and  why  ? 

a.  113.46x0.1111         c.  2137.1x0.0123 

b.  23.684  x  1.6842          d.  1.6931  x  1.2311 

34.  Find  the  product  in  each  of  the  cases  just 
given. 

NOTE.  When  we  multiply  by  a  number  which 
contains  decimal  places,  the  last  step  in  the  process 
always  consists  in  a  division,  the  effect  of  which  is 
to  move  the  point  as  many  places  to  the  left  as 
there  are  decimal  places  in  the  multiplier  ;  we  may 
say,  then,  that  the  number  of  decimal  places  in  the 
product  is  the  sum  of  the  number  of  decimal 
places  in  the  multiplicand  and  in  the  multiplier. 

35.  How  many  decimal  places  are  there  in  the 
product  in  each  of  the  following  cases,  and  why  ? 

a.  68.2 x. 4  j.   186.6 x. 66 

b.  13.45  x. 6  k.  1215  x. 012 

c.  86.2 x. 8  I.   26.943xl.09 

d.  13691  x  .004  m.  .9642  x  .009 

e.  .2841  x. 18  n.  233.1x98 
/.  .1234 x. 11  o.  1684 x. Ill 

g.  1.7236 x. 01  p.  .068x68 

h.  11.123 x. 002  q.  7.123x9.8 

/.  .2783 x. 0123  r.  1.672x1.31 

36.  Find  the  product  in  each  of  the  cases  just 
given. 


B.]  Multiplication.  127 

37.  At  $0.667  a  foot,  what  is  the  cost  of  enough 
steel  rails  for  a  rod  (16.5  ft.),  of  railroad  track  ? 

38.  If  the  circumference    of   a   wheel  is    3.14 
times  the  diameter,  what  is  the  circumference  when 
the  diameter  is  3.87  feet  ? 

39.  If  a  man  can  walk  3.75  miles  in  an  hour, 
how  far,  by  walking  6.3  hours  a  day,  can  he  walk 
in  3.8  days  ? 

40.  If  it  takes  a  train,  which  travels  at  an  aver- 
age rate  of  26.22  miles  an  hour,  10.25  hours  to  go 
from  A  to  B,  what  is  the  distance  between  these 
places  ? 

41.  At  $8.26  an  acre,  what  is  the  cost  of  .06  of 
an  acre  of  land  ?   of  .07  of  an  acre  ?  of  .08  of  an 
acre?     Ans.  10.4956;  10.5782;  $0.6608 

NOTE.  Since  a  cent  is  the  smallest  coin  we 
have  in  circulation,  the  amounts  given  above  can- 
not be  paid  exactly.  In  cases  of  this  kind  the 
number  of  cents  required  is  that  which  is  nearest 
the  exact  amount ;  in  paying  these  amounts,  then, 
there  would  be  required  $0.50  for  the  first,  $0.58 
for  the  second,  and  $0.66  for  the  third. 

42.  What  must  I   pay  for  6.3  square  feet  of 
leather  at  $0.37  a  square  foot?  what  for  8.7  square 
feet?     Ans.  $2.33;  $3.22 

43.  What  must  I  pay  for  9.5   square  feet  of 
leather  at  $0.37  a  square  foot  ?     Ans.  The  exact 
amount  is  $3.515  or  $3.5l£,  which  is  equally  near 
$3.51  and  $3.52      In  a  case  of  this  kind  the  man 
who  sells  generally  adds  a  half  cent  to  his  bill. 


128  Decimals. 

In  this  case,  then,  I  should  probably  be  called  upon 
to  pay  $3.52  * 

44.  What  must  a  carpenter  pay  for  the  follow- 
ing bill  of  lumber:   4500  shingles  at  $4.70  per 
thousand  ;    13,842  feet  of   boards  at  $28.35  per 
thousand;  4849  feet  of  planks  at  $42.75  per  thou- 
sand ;  18,382  laths  at  $0.38  per  hundred  ? 

45.  What  will  18,763  bricks  cost  at  $7.75  per 
thousand  ? 

46.  What   will    be   the    freight   charge   from 
Boston 

(a)  to  Chicago,  on  1263  Ibs.  at  75c.  a  hundred  ? 
(6)  to  New  York,  on  1878  Ibs.  at  35c.  a  hundred? 

(c)  to  Albany,  on  2034  Ibs.  at  30c.  a  hundred  ? 

(d)  to  New  Haven,  on  689  Ibs.  at  28c.  a  hundred  ? 

(e)  to  Buffalo,  on  568  Ibs.  at  44c.  a  hundred? 

*  Where  a  half  cent  comes  into  the  exact  amount,  an  old  busi- 
ness custom  —  still  sometimes  followed  —  gives  the  half  cent  to 
the  man  who  makes  the  change.  If,  for  example,  A  owes  B 
$0.405  and  offers  him  exactly  $0.4(3,  B  should  not  object;  but 
if  A  offers  B  $0.50  then  B  need  give  back  only  3  cents  in 
change. 


SECTION  VI. 
DIVISION  OF  DECIMALS. 

A*  Division  by  a  Whole  Number.    Examples  and 
Problems,  with  Remarks  and  Explanations. 

1.  a.  Divide  9.7  by  4. 

Divisor  4)9.7     Dividend. 
2.4J  Quotient. 

Applying  the  method  used  in  the  division  of 
whole  numbers,  we  say  i  of  9  units  is  2  units,  with 
a  remainder  of  1  unit ;  we  set  down  the  2  in  the 
units'  place  and  save  the  1  [unit].  1  unit  added 
to  7  tenths  makes  17  tenths ;  J  of  17  tenths  is  4J 
tenths  [.4J]  ;  we  set  down  the  .4£  to  the  right  of 
the  2. 

6.  Show  that  9.6  divided  by  4  is  2.4,  by  first  re- 
ducing the  9.6  to  a  common  fraction  and  then 
proceeding  as  in  the  division  of  a  fraction  by  a 
whole  number. 

2.  Divide  9.48  by  6,  in  each  of  the  two  ways 
just  indicated.     A ns.  1.58 

3.  Divide  1.728  by  12,  in  two  ways.   Ans.  0.14^ 

4.  Divide  48.6912  by  24,  in  two  ways. 

5.  a.  How  many  places  are  there  in  the  divi- 
dend, and  how  many  in  the  quotient  in  each  of  the 
last  4  examples  ? 


130  Decimals.  [§  6. 

b.  Give  a  new  example  of  each  of  the  following 
cases  :  With  a  divisor  a  whole  number,  and  a  divi- 
dend containing   one  decimal  place  ;    a  dividend 
with  two  decimal  places  ;    three  decimal  places  ; 
four  decimal  places.     How  many  decimal  places 
will  there  be  in  the  quotient  of  each  ? 

c.  How  many  decimal   places  will  there  be  in 
the  quotient  of  9.68479  divided  by  4  ?     Find  the 
quotient.     Ans.  2.42119| 

6.  Give  a  new  example  of  each  of  the  following 
cases  before  answering  the  question : 

With  the  divisor  a  whole  number,  how  many 
decimal  places  will  there  be  in  the  quotient  when 
the  dividend  contains  one  decimal  place  ?  two 
decimal  places?  three  decimal  places  1  four  deci- 
mal places  ? 

NOTE.  From  what  precedes  we  see  that  when 
the  divisor  is  a  whole  number  we  may  divide  ex- 
actly as  in  whole  numbers,  pointing  off  the  same 
number  of  decimal  places  in  the  quotient  as  are 
pointed  off  in  the  dividend. 

7.  Divide  80.4144  by  4. 

8.  Divide  864.144  by  6. 

9.  Divide  8641.44  by  9. 

10.  Divide  4.216  by  2. 

11.  Divide  0.6824  by  8. 

12.  Divide  0.0132  by  9. 

13.  Divide  6.55  by  7. 

14.  Divide  4.18  by  4. 

15.  Divide  0.167  by  11. 

16.  Divide  3.264  by  24.     Divide  78.428  by  36. 


A.]  Division.  131 

17.  How  many  hundredths  are  there  in  f  (that 
is,  in  3  divided  by  4)  ?     We  know  that  in  3  there 
are  300  hundredths  (?8#),  which  may  be  written 
3.00      Now,  dividing  as  in  previous  examples,  we 

have4)— ,  therefore  | -.75 
.75 

How  many  hundredths  are  there  in  2\  ? 

18.  How  many  thousandths  are  there  in  £  ?     [5 
may  be  written  5.000]     Ans.  .625 

How  many  thousandths  are  there  in  T| ^  ? 

19.  Change  J  to  thousandths.     Ans.  .875 

20.  How  many  tenths  are  there  in  J  ?  how  many 
hundredths  with  how  many  thousandths  left  over  ? 

21.  Reduce  ^  to  hundredths. 

22.  Reduce  f  to  thousandths. 

23.  Reduce  T|5  to  thousandths. 

24.  Reduce  J>J  to  hundredths.     Ans.  .68 
Since  .68  is  nearer  to  .7  than  to  .6,  we  may  say 

that,  expressed  to  the  nearest  tenth,  JJ  is  .7 

25.  What  number  of  tenths  comes  nearest  the 
true  value  of  §£  ?     Ans.  .8 

26.  What  number  of  tenths  comes  nearest  the 
true  value  of  J  ?  (Ans.  .3)  What  number  of  hun- 
dredths? (Ans.  .33) 

27.  Express  to  the  nearest  hundredth  the  value 
of  §,  of  f ,  of  ft,  of  T\. 

28.  a.  How  many  hundred-thousandths  are  there 
in  ,  ?    7)2.00000   ' 

0.28571+  Ans. 

[The  +  is  used  here  to  denote  that  there  is  still 
a  remainder  after  using  five  places.] 


132  Decimals.  [§  6. 

1).  How  many  In u id  red-thousandths  are  there  in  |  ? 

&  =  . 77777,  etc.  Using  only  4  decimal  places  we 
should  call  our  answer  .7778,*  because  this  is 
nearer  the  true  value  of  J  than  .7777  In  this 
case  we  may  say,  if  we  like,  ^  =  .7778  — 

29.  How  many  hundredths  are  there  in  {  ? 
Since  j  =.625  is  just  half  way  between  .63  and 

.62,  we  may  here  choose  for  our  answer  either  .62 
or  .63  Where  there  is  any  choice,  most  com- 
puters, for  the  sake  of  uniformity,  choose  an  even 
number  for  the  last  figure.  They  would  therefore 
choose  .62  rather  than  .63 

30.  How  many  hundredths  are  there  in  Jg  ? 

.I//*.  .82 

31.  Divide  11  by  13,  carrying  out  the  result  to 
4  decimal  places  only.     The  work  may  be  written 
as  indicated  below  on  the  left ;  or,  since  we  can 
easily  tell  how  many  zeros  are  to  be  used  without 
actually  writing  them  all  in  the  dividend,  it  may 
be  written  more  briefly  as  indicated  on  the  right. 
13)11.0000(.8461+      13)11.0(.8461  + 

104  104 

60  ~~60 

52  52 


80  80 

78  78 

~20  ~20 

13  13 

*  The  result  should  always  be  made  as  near  the  true  value  as 
possible. 


A.]  Division.  133 

32.  Divide  11  by  14,  carrying  out  the  result  to 
3  places  of  decimals.     Ans.  .786 

33.  Divide  (a)   23  by  32,  and  (6)  87  by  64, 
continuing  the  process  of  division  in  each  case  un- 
til there  is  no  remainder. 

34.  Divide  18  by  12.     Ans.  1.5 

35.  Divide  25  by  8.     Ans.  3.125 

36.  Divide  27  by  24. 

37.  Divide  11  by  8. 

38.  Divide  8  by  11,  carrying  out  the  result  to 
three  places  of  decimals.     Ans.  0.727 

39.  Divide  2  by  9,  carrying  out  the  result  to  3 
places  of  decimals. 

40.  Divide  9  by  24. 

41.  An  Englishman  on  reaching  America  sold 
3  English  pounds  for  $  13.58  ;  how  much  was  that 
a  pound  ? 

42.  Change  1.3  gills  to  the  decimal  of  a  pint. 
[1  gill  =  i  of  a  pint,  therefore  1.3  gills  =  lf-  pts.  = 
.325  pts.] 

43.  Change  1  gill  to  the  decimal  of  a  quart. 

44.  Change  3  gills  to  the  decimal  of  a  gallon. 

Ans.  .09375  galls. 

45.  Change   2  qts.   1   pt.   to  the  decimal  of  a 
gallon. 

Solution:  First  step,  1  pt.  =  .5  qts.    .*.  2  qts.  1  pt. 
=  2.5  qts.    Second  step,  2.5  qts.  =  .625  galls.   Ans. 
The  work  may  be  arranged  as  follows  : 
2  1.0  pt. 


2.5  qts. 
.625  galls.  Ans. 


134  Decimals.  [§  6. 

46.  Change  3  qts.  1  pt.  2  gi.  to  the  decimal  of 
a  gallon. 


Solution:  4 


2.0  gi. 


1.50  pts. 


3.75  qts. 


.9375  galls.  Ans. 

47.  Change  1  pt.   1.6  gi.  to  the  decimal  of  a 
gallon. 

48.  Change  1  qt.  1  pt.  3.52  gi.  to  the  decimal 
of  a  gallon. 

49.  Change  1  pt.  2.96  gi.  to  the  decimal  of  a 
gallon. 

50.  If  2  quarts  of  alcohol  cost  $2.16,  what  will 
be  the  cost  of  a  gill  ? 

51.  If  24  gallons  of  vinegar  cost  16.72,  what 
will  be  the  cost  of  a  pint  ? 

52.  If   8   two-gallon  cans  of  milk  cost  #6.72, 
what  will  be  the  cost  of  a  pint  ? 

53.  If  3  gallons  of  ice  cream  cost  $ 4.80,  what 
will  be  the  cost  of  a  pint  ? 

54.  If  4  gallons  of  molasses  cost  $ 2.00,  what 
will  be  the  cost  of  a  quart  ? 

55.  If  3  gallons  of  astral  oil  cost  75  cents,  what 
will  be  the  cost  of  a  quart  ? 

56.  Change  1  pk.  3  qts.  0.4  pts.  to  the  decimal 
of  a  bushel.     Ans.  .35  bu. 

57.  What  is  the  number  of  bushels  expressed 
decimally  in  1  bu.  1  pk.  6  qts.  .8  pts.  ? 

58.  Change  3  qts.   1  pt.  to  the  decimal  of  a 
bushel. 

59.  Change  3  pks.  7  qts.  1  pt.  to  the  decimal  of 
a  bushel. 


A.]  Division.  135 

60.  Change  1.28  pts.  to  the  decimal  of  a  bushel. 

61.  If  3  pecks  of  berries  cost  12.88,  what  will  a 
quart  cost  ? 

62.  If  a  bushel  of  grass  seed  costs  $4.64,  what 
will  a  quart  cost  ? 

63.  If  2  bushels  of  potatoes  cost  11.50,  what  will 
a  peck  cost  ? 

64.  If  3  bushels  of  apples  cost  13.42,  what  will 
a  peck  cost? 

65.  Change  32  ounces  to  the  decimal  of  a  ton. 

66.  Change  5  pounds  to  the  decimal  of  a  ton. 

67.  Change  1624  Ibs.  12  oz.  to  the  decimal  of  a 
ton. 

68.  Change  1  Ib.  3.2  oz.  to  the  decimal  of  a  ton. 

69.  Change  430  Ibs.  3.84  oz.  to  the  decimal  of 
a  ton. 

70.  If  14  pounds  of  coffee  cost  $5.25,  how  much 
will  a  pound  cost  ? 

71.  If  a  ton  of  range  coal  costs  $5.50,  what  will 
a  pound  cost  ? 

72.  If  a  ton  of  American  Cannel  coal  costs  $12, 
what  will  a  pound  cost  ? 

73.  If  a  ton  of  hay  costs  $22,  what  will  a  pound 
cost? 

74.  If  3  tons  of  lead  cost  $690,  what  will  a 
pound  cost  ? 

75.  Change  63.36  inches  to  the  decimal  of  a 
mile. 

Solution :  We  may  change  successively  to  feet, 
yards,  rods,  and  miles,  and  may  arrange  our  work 
as  follows : 


136  Decimals.  [§  6. 

12)03.86  in. 
3)5.28  ft. 

here  dm^"  by  H  and 


*i_  ix 

°2  ~  *~        .16  multiply  by  2. 

320)^20  rds. 

.001  mi.  Ans. 

76.  Change  40  rds.  1  ft.  3.84  in.  to  the  decimal 
of  a  mile.     Ans.  .12525  miles. 

77.  Change  3  yds.  1  ft.  6.72  in.  to  the  decimal 
of  a  mile. 

78.  Change  36  rds.  1  yd.  2  ft.  3.36  in.  to  the 
decimal  of  a  mile. 

79.  If  it  costs  $11.59  to  build  38  rods  of  fence, 
what  will  it  cost  to  build  a  rod  ? 

80.  If  108  fathoms  of  rope  cost  18.10,  what  will 
a  fathom  cost  ?     [A  fathom  =  6  ft.] 

81.  How  much   is   railroad    iron    a  foot  when 
rails  enough  to  lay  a  mile  of  track  cost  f  580.80  ? 

82.  If  it  takes  sound  42.24  seconds  to  travel  10 
miles,  how  long  will  it  be  in  going  1  foot  ?     How' 
many  feet  will  it  travel  in  a  second  ? 

83.  Change  66  rds.  4  yds.  1  ft.  2.4  in.  to  the 
decimal  of  a  mile. 

84.  Change  36  rds.  4  yds.  1  ft.  2.4  in.  to  the 
decimal  of  a  mile. 

85.  Change  3  rds.  3  yds.  2  ft.  3.432  in.  to  the 
decimal  of  a  mile. 

86.  Change  1   rd.  1  ft.  3.84  in.  to  the  decimal 
of  a  mile. 

87.  Change  to  the  decimal  of  a  mile  33  hands; 
52.8  hands.     [A  hand  =  4  in.] 


I 


B.]  Division.  13' 

88.  Change  36  seconds  to  the  decimal  of   an 
hour. 

89.  Change  1  h.  12  min.  to  the  decimal  of  a 
day. 

90.  Change  15  h.  39  min.  36  sec.  to  the  decimal 
of  a  day.     Ans.  .6525  dys. 

91.  Change  4  h.  22  min.  49.08  sec.  to  the  deci- 
mal of  a  day.     Ans.  .1825125  dys. 

92.  Change  3  h.  30  min.  23.04  sec.  to  the  deci- 
mal of  a  day. 

93.  Change  2  h.  15  m.  to  the  decimal  of  a  week. 


B.  Division  of  a  Decimal  by  a  Decimal.  Exam- 
ples and  Problems,  with  Hemarks  and  Expla- 
nations. 

1.  Divide  86.4144  by  1.2 

Since  1.2  is  the  same  as   \\  we  may  say  that 
86.4144 -M.2- 86.4144^ \\.     Now  to  divide  by  1$ 
we  may  first  dmde^by  12  and  then  multiply  by 
10  ;  we  know  therefore  that         , 
86.4144-7-13 -=  ^\\ 4 4  x  10 :  =  7.20^2  x  10- 72.012 

2.  Show  that  86.4144 -f. 6  =  144.024^ 

3.  Show  that  86.4144^-2.4  =  36.006 

4.  Show  that  86.4144-f-.9-96.016 

5.  Divide  86.4144  by  .8 

Here,  as  in  the  last  four  examples,  two  steps  are 
to  be  taken :  First,  we  are  to  divide  by  8,  and, 
second,  we  are  to  multiply  by  10.  Now  multiply- 
ing by  10  moves  the  point  one  place  to  the  right 


138  Decimals.  [§  6. 

(see  p.  Ill),  and  therefore  gives  only  3  decimal 
places  in  the  quotient,  or  one  less  than  there  are 
in  the  dividend.  We  may,  then,  arrange  our  work 
as  follows,  first  dividing  by  8  as  in  whole  numbers, 
and  then  pointing  off  only  3  decimal  places  in  the 
quotient. 

Divisor  0.8)86.4144  Dividend. 
108.018  Quotient. 

6.  How  many  decimal  places  are  there  in  the 
quotient  in  each  of  the  following  examples,  and 
why? 

a.  4.263^.3      c.  6.1224^-.^.  5.064^-2.4 
6.  91.26-^.9       d.  8.8-5-  .8      /  0.01728^14.4 

7.  Find  the  quotient  in  6  a. 

8.  Find  the  quotient  in  6  b. 

9.  Find  the  quotient  in  6  c. 

10.  Find  the  quotient  in  6  d. 

11.  Find  the  quotient  in  6  e. 

12.  Find  the  quotient  in  6/1 

13.  Divide  86.4144  by  .06 

Since  .06  is  the  same  as  T$Q,  we  may  say  that 


86.4144  *  .06  =  86.4144  *  T-  = 

14.4024  x  100  -  1440.24  Ans. 

How  does  this  quotient  differ  from  that  of  Ex- 
ample 2  ? 

14.  Show  that  86.4144^0.12  =  720.12 
Compare  with  Example  1. 

15.  Show  that  86.4144-K24-  360.06 
Compare  with  Example  3. 

16.  Show  that  86.4144  -r.  09  =  960.16 
Compare  with  Example  4. 


B.]  Division.  139 

17.  Divide  86.4144  by  .08 

Here  the  two  steps  in  our  work  are :  (1)  to  di- 
vide by  8  and  (2)  to  multiply  by  100.  Now  mul- 
tiplying by  100  moves  the  point  two  places  to  the 
right  (see  p.  112),  and  therefore  gives  only  two 
decimal  places  in  the  quotient,  or  two  less  than 
there  are  in  the  dividend.  We  may,  then,  arrange 
the  work  as  follows,  first  dividing  by  8  as  in 
whole  numbers,  and  then  pointing  off  only  2  deci- 
mal places  in  the  quotient. 

.08)86.4144 
1080.18 

Compare  with  Example  5. 

18.  How  many  decimal  places  are  there  in  the 
quotient  in  each  of  the  following  examples,  and 
why? 

a.  4.263^.03    c.  6.1224^.04    e.  5.064 -r. 24 

b.  91.26-K09    d.  .88-r.OS        /.  .01728-^1.44 

19.  Find  the  quotient  in  18  a. 

20.  Find  the  quotient  in  18  b. 

21.  Find  the  quotient  in  18  c. 

22.  Find  the  quotient  in  18  d. 

23.  Find  the  quotient  in  18  e. 

24.  Find  the  quotient  in  18  f. 

25.  Divide  86.4144  by  .006 

Since  .006  is  the  same  as  T^-Q  we  may  say  that 
86.4144-^.006  =  86.4144^^  =  86.4144  x  1000  ^ 

14.4024x1000  =  14402.4.  How  does  this  quotient 
differ  from  those  of  Examples  2  and  13  ? 

26.  Show  that  86.4144 -r. 012  =  7201.2 


140  Decimals.  [§  6. 

Compare  with  Examples  1  and  14. 

27.  Show  that  86.4144 ^.024 -3600.6 
Compare  with  Examples  3  and  15. 

28.  Show  that  86.4144-- .009  =  9601.6 
Compare  with  Examples  4  and  16. 

29.  How  many  decimal  places  are  there  in  the 
quotient   of  86.4144  divided   by  .008,  and   why  ? 
Find  the  quotient  by  arranging  the  work   as  in 
Examples  5  and  17. 

30.  How  many  decimal  places  are  there  in  the 
quotient  in  each  of  the  following  examples,  and 
why  ? 

a.  4.263-^.003  d.  .00088^.008 

6.  .9126^.009  e.  5.064 -r. 024 

c.  6.1224^.004  /.  .01728-T.144 

31.  Find  the  quotient  in  30  a. 

32.  Find  the  quotient  in  30  6. 

33.  Find  the  quotient  in  30  c. 

34.  Find  the  quotient  in  30  d. 

35.  Find  the  quotient  in  30  e. 

36.  Find  the  quotient  in  30 /. 

37.  When  the  divisor  contains  3  decimal  places 
(thousandths)  the  number  of  decimal  places  in  the 
quotient  will  be  how  many  less  than  in  the  divi- 
dend, and  why? 

NOTE.  In  each  of  the  last  37  examples  (where 
the  divisor  contains  not  more  than  three  decimal 
places)  the  number  of  decimal  places  in  the  quo- 
tient was  found  to  be  equal  to  the  difference  be- 
tween the  number  of  decimal  places  in  the  divi- 
dend and  in  the  divisor.  That  this  statement  is 


B.]  Division.  141 

true,  for  any  example  where  the  divisor  has  more 
than  three  decimal  places  can  easily  be  shown  by 
reducing  the  divisor  to  a  common  fraction,  as  in 
Example  1. 

38.  How  many  decimal  places  are  there  in  the 
quotient  of  0.643214  divided  by  0.00011?     Find 
the  quotient. 

39.  How  many  decimal  places  are  there  in  the 
quotient  of  0.001233  divided  by  .0003  ?     Find  the 
quotient,  arranging  the  work  as  in  Example  17. 

40.  Why  are  there  no  decimal    places  in  the 
quotient  of  .0066  divided  by  .0011?     Find  the 
quotient. 

[The  difference  between  the  number  of  decimal 
places  in  the  dividend  and  in  the  divisor  being  zero, 
the  statement  contained  in  the  note  following  Ex- 
ample 37  holds  good  as  well  here  as  in  previous 
examples.] 

41.  a.  Divide  5.7102  by  .1842 

.1842)5.7102(31.  Answer. 
5526 
1842 
1842 

6.  Divide  0.01664  by  .0064.     Ans.  2.6 

42.  Divide  .697368   by  .000168      Ans.  4151 

43.  Divide  6.316125  by  1.6843      Ans.  3.75 

44.  Divide  16.7376  by  1.32;  8.552478  by  1.111 

45.  Divid^.199584  by  2.31;  .625  by  .25 

46.  Divide  .6  by  .06 

The  statement  contained  in  the  note  following 


142  Decimals.  [§  6. 

Example  37  does  not  appear  to  hold  good  here  be- 
cause the  number  of  decimal  places  in  the  divi- 
dend is  less  than  in  the  divisor ;  we  will  therefore 
first  solve  this  example  by  the  method  of  Exam- 
ples 1,  13,  and  25,  where  the  divisor  was  in  each 
case  reduced  to  a  common  fraction  : 

.6-K06  =  .6-J-Tfa  =  .6x  100  =  .1x100  =  10. 

6 

47.  Divide  .044  by  .0011 

48.  Divide  12.33  by  .009 

49.  Divide  14.4  by  .012 

50.  Although  the  method  given  above  offers  no 
difficulty,  we  may  find  the  quotients  by  a  shorter 
process,  as  follows : 

a.  In  Example  46  the  dividend  .6  or  -f$  is  the 
same  as  T\0^,  and  may  be  written  .60     We  can 
now  divide  .60  by  .06  as  in  Example  41  a,  thus : 
.06).6Q 

10. 

b.  In  Example  47   the  dividend  .044   may  be 
written  .0440 ;   «  may,  then, 

divide    as   indicated    on    the  '  — ^~ 

right,  and   place  the  decimal 
point  in  the  quotient  in  accordance  with  the  state- 
ment contained  in  the  note  on  page  140. 

c.  Solve  in  the  same  way  Example  48. 

51.  Solve  in  the  same  way  Example  49. 

52.  From  the  last  six  examples  we  learn  that 
the  statement  contained  in  the  notion  page  140 
can   be  made  to  apply  to   cases  where  there  are 
fewer  decimal  places  in  the  dividend  than  in  the 


C.]       Reduction  of  Common  Fractions.         143 

divisor,  if  we  first  annex  zeros  to  the  dividend 
until  it  contains  as  many  decimal  places  as  there 
are  in  the  divisor. 

Divide  172.8  by  .144     Answer  1200 

53.  Divide  1.8  by  .006 

54.  Divide  .144  by  .0004 

55.  Divide  86  by  .43 

56.  Divide  2.8  by  .007 

57.  Divide  1  by  .25 

58.  Divide  10  by  .02 

59.  Divide  50  by  .001 


C.  Reduction  of  Common  Fractions  to  Deci- 
mals.    Circulating  Decimals. 

Some  common  fractions  can  be  exactly  expressed 
by  decimals,  and  others  cannot  be  so  expressed. 
In  the  following  cases,  for  example,  each  fraction 
can  be  exactly  expressed  by  a  decimal : 

|  =.75       $=.625       ^=.35         T|^  =  -072 
2'r,  =  .16       $=.875        |  =.375  _    7^  =  .008 
But  if  we  try  to  reduce  ^  to  a  decimal,  each  suc- 
cessive division  gives  a  quotient     Q^  QQQOOO 
3  and  a  remainder  1,  and  this         -•  OOOOOQI  _  i 
will  evidently  continue  to  be  the 
case  no  matter  how  far  the  process  of  division  be 
continued.      There  can  be,  therefore,  no  decimal 
fraction  which  is  the  exact  equivalent  of  ^.     The 
decimal  .3  expresses  the  value  of  J  to  within  ^; 
.33  to  within  T^ ;  .3333  to  within  ToiroT,  etc' 


144  Decimals.  [§  G. 

1.  Find  the  decimal  of  three  places  which  ex- 
presses to  within  -^-^  the  value  of  |.     Ans.  .666 
[Expressed  to  the  nearest  thousandth  the  answer 
would  be  .667] 

What  figure  would  be  repeated  if  the  process  of 
division  were  continued  ? 

2.  Find  the  decimals  which  express  to  within 
TtfFoT)  the  values  of  the  following  fractions  :   («) 

4;  (*)|;  001;  001;  00  I ;  (/)  5- 

What  figure  would  be  repeated  in  each  case  if 
the  process  of  division  were  continued  ? 

3.  Find  the  decimals  which  express  to  within 

nnfWlF  the  fractions  (a)  ^  ;  (&)  T^  ;  0)  1 .1- 

What  figure  would  be  repeated  in  each  case  if 
the  process  of  division  were  continued  ? 

4.  Find  the  decimals  which  express  to  within 
Ttfowoo  the  fractions,  (a)  ff  ;  (6)  f f  ;  (c)  ,;!,r 
[Answer  to  (a)  .696969  +  ] 

What  ^o  figures  would  be  repeated  in  each 
case  if  the  process  of  division  were  continued  ? 

5.  Find  the  decimals  which  express  to  within 

TOO^OOIT  («)  HI ;  (6)  III-     ^n«.  (a)  .684684  + ; 
(6)  .362362  + 

What  three  figures  would  be  repeated  in  each 
case  if  the  process  of  division  were  continued  ? 

6.  Divide  1  by  7,  carrying  out  the  result  to  12 
places  of  decimals.     Ans.  .142857142857  + 

What  six  figures  would  be  repeated  if  the  pro- 
cess of  division  were  repeated  ? 

7.  Divide  6  by  13,  carrying  out  the  result  to  12 
places  of  decimals.     Ans.  .461538461538  + 


D.]  Miscellaneous  Examples.  145 

What  six  figures  would  be  repeated  if  the  pro- 
cess of  division  were  continued  ? 

NOTE.  In  the  last  seven  examples  we  have  dealt 
with  decimals  in  which,  if  the  process  of  division 
were  continued,  the  same  figure  or  series  of  figures 
would  be  repeated,  over  and  over  again ;  such  deci- 
mals are  called  CIRCULATING  DECIMALS,  and  are 
more  commonly  written  by  placing  a  dot  over  the 
first  and  last  of  the  series  of  repeated  figures. 

Thus,  in  Example  4,  J|  =  .696969  +  may  be  more 

briefly  written  ff  =.69 

.1428571428574-  (see  Example  6)  may  be  writ- 
ten .142857 

.362362  4-  (see  Example  5  6)  may  be  written 
.362 

.58333  -f    (see  Example   3  6)    may  be   written 

.583;  .666+  (see  Example  1)  may  be  written  .6 
Express  in  the  same  manner  the  remaining   an- 
swers of  the  last  seven  examples. 


D.  Miscellaneous  Examples. 

1.  If  a  man  walks  at  the  rate  of  a  mile  in  .4  of 
an  hour,  how  far  can  he  walk  in  3.432  hours  ? 

2.  In  one  rod  there  are  5.5  yards  ;  how  many 
rods  are  there  in  27.225  yards? 

3.  The  quart  of  liquid  measure  contains  57.75 
cubic  inches  ;  how  many  quarts  of  liquid  measure 
are  there  in  a  cubic  foot   (1728  cubic  inches)  ? 
Carry  the  result  out  to  two  places  of  decimals. 


146 


Decimals. 


[§6. 


4.  The  quart  of  dry  measure  contains  67.2  cu- 
bic inches ;  how  many  quarts  of  dry  measure  are 
there  in  a  cubic  foot?      Carry  the  result  out  to 
two  places  of  decimals. 

5.  At  $7.75    per  thousand,  how    many  bricks 
can  be  bought  for  $26.35  ? 

6.  At  35  cents  a  hundred,  how  many  laths  can 
be  bought  for  183.86  ? 

7.  If  in  34  lines  there  are  331  words,  what  is 
the  average  number  of  words  per  line  ?     Carry 
out  the  result  to  two  places  of  decimals. 

Ans.   9.74 

8.  The  figures  given  below  are  taken  from  the 
Census  of  1880.     Compute  to  two  places  of  deci- 
mals the  average  size  in  acres  and  the  average 
value  of  the  farms  of  each  state. 


Number  of 
farms. 

No.  of  acres 
in  farms. 

Value  of  farms. 

Alabama 

135,864 

6,375,706 

$  78,954,648 

California 

35,934 

10,669,698 

262,051,282 

Colorado 

4,506       616,169 

25,109,223 

Massachusetts 

38,406    2,128,311 

146,197,415 

Michigan 

154,008    8,296,862 

499,103,181 

Ohio 

247,18918,081,091 

1,127,497,353 

Pennsylvania 

213,542  13,423,007 

975,689,410 

South  Carolina 

93,864    4,132,050 

68,677,482 

Texas 

174,18412,650,314 

170,468,886 

Virginia 

118,517 

8,510,113 

216,028,107 

9.  In  1887  the  Chicago,  Burlington,  and  Quincy 
Railroad  Company  sold 


D.]  Miscellaneous  Examples.  147 

a.  In   Nebraska    7,079.48   acres   of    land    for 
151,739.58 

b.  In  Iowa  7,357.32  acres  of  land  for  151,724.11 
What  was  the  average  price  per  acre  in  each 

case  ?     Carry  out  the  result  to  two  places  of  deci- 
mals. 


SECTION  VII. 
PERCENTAGE. 

A.  Interest. 

1.  If  I  must  pay  6  cents  for  the  use  of  $1  for  1 
Money          year,  what  must  I  pay  for  the  use  of  $ 6  for 

theYse'of     the  same  time  ?  what  for  the  use  of  $16  ? 
money.  of  $8  ?  of  *  10.50  ?  of  118.75  ?  of  $25  ? 

2.  If  I  must  pay  6  cents  for  the  use  of  $1  for 

1  year,  what  must  I  pay  for  the  use  of  $  1   for 

2  years?   what  for  6  years?   for  2  years  and  6 
months?*  for  4  years  and  4  months  ?  for  10  years 
and  4  months  ?    for  3  years  and  8  months  ?   for 
2  months  ?    for  4  months  ?   for  6  months  ?  for  8 
months?    for  10  months?    for   1   month?    for   3 
months?  for  9  months? 

3.  a.  If  I  must  pay  8  cents  for  the  use  of  $1 
for  1  year,  how  much  must  I  pay  for  the  use  of  f  3 
for  4  years? 

Solution :  If  I  must  pay  8  cents  for  the  use  of 
f  1  for  1  year,  for  the  use  of  $3  for  1  year  I  must 
pay  3  times  8  cents,  or  24  cents. 

If  I  must  pay  24  cents  for  the  use  of  $3  for  1 

*  In  cases  of  this  kind,  business  men,  for  the  sake  of  conven- 
ience, usually  regard  a  month  as  ^  of  a  year:  this,  though  very 
nearly  true,  is  not  exactly  so,  since  no  month  contains  a  number 
of  days  which  is  exactly  ^  of  365. 


A.]  Interest.  149 

year,  for  the  use  of  $3  for  4  years  I  must  pay  4 
times  24  cents,  or  96  cents. 

b.  If  I  must  pay  8  cents  for  the  use  of  11  for 
1  year,  how  much  must  I  pay  for  the  use  of  $6  for 
7  years?  Ans.  $3.36.  How  much  for  the  use  of 
f3.50  for  3  years  ?  Ans.  84  cents.  How  much  for 
the  use  of  $6.50  for  5  years  and  6  months?  Ans. 
$2.86.  How  much  for  the  use  of  $1  for  6  months? 
for  3  months  ?  9  months  ?  1  month  ? 

4.  If  I  must  pay  4  cents  for  the  use  of  $1  for 
1  year,  what  must  I  pay  for  the  use  of  $8  for  3 
years  ?  what  for  the  use  of  $7  for  6  years  and  3 
months  ?  what  for  the  use  of  $10.50  for  5  years 
and  6  months  ?  what  for  the  use  of  $2  for  12  years 
and  9  months  ? 

5.  If  I  must  pay  $6  for  the  use  of  $100  for  1 
year,  how  much  must  I  pay  for  the  use  of  $300  for 
4  years  ?  $400  for  2  years  ?  $200  for  3  years  and 
6  months  ?  $500  for  3  years  and  2  months  ?  $600 
for  1  year  and  9  months  ?  $100  for  2  years  and  8 
months  ? 

6.  If  an  Englishman  has  to  pay  T|^  of  IX  for 
the  use  of  1£  for  1  year,  how  many  shillings  must 
he  pay  for  the  use  of  3£  for  4  years?  (Ans.  12s.) 
5£  for  3  years  ?  4X  for  2  years  and  6  months  ? 
10<£  for  6  months  ?  9X  for  4  months  ? 

7.  If  I   must  pay  $6  for  the  use  of  $100  for  1 
year,  what  must  I  pay  for  the  use  of  $100  for  1 
month?  for  15  days?  *  for  6  days?  for  24  days? 

*  In  cases  of  this  kind,  business  men  usually  regard  1  MONTH 
AS  30  DAYS,  and  therefore  I  DAY  AS  ^  OF  A  MONTH,  15  days  as 

io  or  i  °*  a  montn- 


150  Percentage.  [§  7. 

for  18  days?  for  12  days?  for  3  days?  for  1  day? 
for  8  days  ? 

8.  If  I  must  pay  $8  for  the  use  of  $100  for  1 
year,  what  must  I  pay  for  its  use  for  1  month  ?  for 
15  days?  for  20  days?  for  10  days?  for  16  days? 

Money  paid  by  a  borrower  to  a  lender  for  the 
interest,  use  of  money  is  called  INTEREST. 

9.  If  the  interest  on  $  1  for  1  year  is  6  cents, 
what  is  the  interest  on  $3.50  for  4  years  and  2 
months  ?     Ans.  $0.875 

10.  If  the  interest  on  $3  for  4  years  is  72  cents, 
what  is  the  interest  on  $1  for  1  year  ?   Ans.  $0.06 
(6  cents). 

11.  If  the  interest  on  $4  for  3  yrs.  4  m.  is  80 
cents,  what  is  the  interest  on  $1  for  1  yr.  ? 

Solution :  If  the  interest  on  $4  for  3  yrs.  4  m. 
is  80  cents,  the  interest  on  $1  for  the  same  time  is 
|  of  80  cents,  or  20  cents.  3  yrs.  4  m.  is  the  same 
as  3^  or  -^-  yrs. ;  now  if  the  interest  on  $1  for  JgQ- 
yrs.  is  20  cents,  then  for  1  of  a  year  it  is  T^  of  20 
cents,  or  2  cents,  and  for  a  whole  year  it  is  3  times 
2  cents,  or  6  cents. 

12.  If  the  interest  on  $1  for  1  yr.  is  6  cents,  what 
is  the  interest  on  $0.50  for  10  yrs.  ?  for  24  days  ? 

13.  If  the  interest  on  $100  for  1  yr.  is  $6,  what 
is  the  interest  on  $300  for  3  yrs.  6m.? 

14.  If  the  interest  on  $1000  for  1  yr.  is  $60, 
what  is  the  interest  on  $100  for  the  same  time? 
on  $1? 

The  usual  interest  for  1  yr.  is  6  cents  on  each 
Per  cent,  dollar,  6  dollars  on  each  hundred  dollars, 


A.]  Interest.  151 

or  Y^Q-  of  the  sum  borrowed ;  the  rate  of  interest 
is  here  said  to  be  6  PER  CENT  *  a  year ;  3  per  cent, 
4  per  cent,  etc.,  signify,  then,  T|T,  T^,  etc.,  of  the 
sum  borrowed. 

15.  At  6  per  cent  a  year,  what  is  the  interest 
on  |3  for  4  yrs.  ?   (Ans.  72  cents)  on  |2  for  6 
yrs.  ?  on  $10  for  2  yrs.  ?  on  $50  for  4  yrs.  8  m. 
6  d.  ?  (Ans.  $14.05) 

16.  At  4  per  cent  a  year,  what  is  the  interest 
on  10<£  for  3  yrs.  ?  (Ans.  1£  4s.)  on  160£  for  5 
yrs.  9  m.  ?  (Ans.  36£  16s.) 

17.  At  8  per  cent  a  year,  what  is  the  interest 
on  $4  for  2  yrs.  3m.? 

18.  At  3  per  cent  a  year,  what  is  the  interest 
on  $6  for  8  yrs.  8m.?' 

19.  At  6  per  cent  a  year,  what  is  the  interest 
on  $12.25  for  3  yrs.  8m.? 

Instead  of  the  words  per  cent,  the  symbol  °/o  is 
often  used  ;  instead,  then,  of  6  per  cent, 
4  per  cent,  8  per  cent,  etc.,  we  may  write  /& 

6$fc,  4£>,  8J6,  etc. 

20.  At  6fi  a  year,  what  is  the  interest  on  $1  for 
2  months  ?  for  4  months  ?  for  6   months  ?  for  8 
months  ?  for  10  months  ?  for  1  month  ?  for  £  of  a 
month  ?  for  18  days  ? 

21.  At  8fi  a  year,  what  is  the  interest  on  $1 
for  3m.?  for  6m.?  for  9m.?  for  15  d.  ? 

22.  At  6/0  a  year,  how  long  will  it  take  $1  to 
earn  12  cents  ?  how  long  to  earn  3  cents  ?  2  cents  ? 

*  Per  cent  is  from  the  Latin  per  centum,  which  means  by  the 
hundred. 


152  Percentage.  [§  7. 

4   cents  ?  1    cent  ?  5  mills  or  |-  of  a  cent  ?  100 
cents  or  $1  ? 

23.  At  8^>  a  year,  how  long  will  it  take  1100  to 
earn  $8  ?  how  long  to  earn  $2  ?  14  ?  $1  ?  $24  ? 
$100? 

24.  At  6<fc  a  year,  how  long  will  it  take  $1000 
to  earn  $60  ?  how  long  to  earn  $10  ?  $5  ?  $20  ? 
$30  ?  $2  ?  $1  ? 

25.  At  6^>  a  year,  how  long  will  it  take  $100  to 
earn  $1  ?  how  long  to  earn  $2.50  ?  $4.50  ?  $1.50  ? 
$5.50  ?  50  cents  ?  10  cents  ?  5  cents? 

NOTE.  Since  interest  is  usually  reckoned  by 
the  year,  6jfc,  4^b,  etc.,  when  not  followed  by  any 
specified  time  are  understood  to  mean  6jfc  a  year, 
4fy  a  year,  etc. 

26.  If  the  interest  on  $2  for  5  yrs.  is  90  cents, 
what  is  the  rate  per  cent.  ?    Ans.  9$fc. 

27.  What  is  the  rate  per  cent  when  the  interest 
on  $1  for  3  yrs.  is  18  cents  ?  what  when  the  inter- 
est on  $1  for  2  in.  is  1  cent  ?  when  the  interest  on 
$1  for  3  yrs.  9  m.  is  30  cents  ?  what  when  the  in- 
terest on  $1  for  2  yrs.   6   m.   is   10  cents  ?  what 
when  the  interest  on  $1  for  6  days  is  1  mill  ? 

28.  What  is  the  rate  per  cent  when  the  interest 
on  $1000  for  1  yr.  is  $40  ?  what  when  the  interest 
on  $1000  for  1  yr.  is  $60  ?  what  when  the  interest 
on  $1000  for  1  yr.  is  $70  ? 

29.  What  is  the  rate  per  cent  when  the  interest 
on  !<£  for  1  yr.  is   Is.  ?  when  the  interest  on  3X 
for  1  m.  is  3d.  ? 

30.  What  is  the  rate  per  cent  when  the  interest 
011  $300  for  2  yrs.  6  m.  is  $30  ? 


A.]  Interest.  153 

31.  What  is  the  rate  per  cent  when  the  interest 
on  $200  for  3  yrs.  6  m.  is  $49  ?  when  the  interest 
on  the  same  sum  for  the  same  time  is  f  42  ?    ' 

32.  What  is  the  rate  per  cent  when  the  interest 
on  $12  for  3  m.  is  12  cents  ? 

33.  If  I  borrow  $100  for  6  m.  3  d.  at  6^,  how 
much  must  I  pay  back,  including  the  $100  and  in- 
terest, at  the  end  of  the  6  months  and  3  days  ? 

34.  If  I  borrow  $20  for  3  yrs.  6  m.  at  8^,  how 
much  must  I  pay  back  ? 

The   sum   borrowed  is  called  the  PRINCIPAL, 
and   the  total  sum  paid  back,  includ-     principal. 
ing    the    Principal    and    Interest,    is     Amount. 
called  the  AMOUNT. 

35.  If  the  principal  is  $50,  what  will  be  the 
amount  at  the  end  of  6  years,  when  the  rate  of 
interest  is  5j&  ? 

36.  If  the  principal  is  $7000  what  will  be  the 
amount  at  the  end  of  10  yrs.  3  m.,  when  the  rate 
of  interest  is  4^  ? 

37.  What  principal,  if  put  at  interest  at  8*fe, 
will  earn  $16  in  two  years  ? 

38.  What  principal,   if  put  at  interest  at  6^, 
will  earn  $1.50  in  8  yrs.  4  m. 

Solution :  $1  will  earn  50  cents  in  8  yrs.  4  m. 
at  Qfi  ;  therefore,  to  earn  $1.50  will  take  as  many 
times  one  dollar  as  50  cents  is  contained  in  $1.50, 
or  3  times  one  dollar.  The  answer,  then,  is  $3. 

39.  What  principal,  if  put  at  interest  at  6^, 
will  earn   $25  in  8  yrs.  4m.?     [Find  first  how 
much  $1  will  earn  in  the  given  time.] 


154  Percentage.  [§  7. 

40.  What  principal,  if   put  at  interest  at  4j6, 
will  earn  60  cents  in  7  yrs.  6m.? 

41.  What  is  the  principal  when  the  amount  at 
the  end  of  1  yr.,  at  6^,  is  1106  ? 

42.  What  is  the  principal  when  the  amount  at 
the  end  of  2  yrs.,  at  4^,  is  $108  ? 

43.  What  is  the  principal  when  the  amount  at 
the  end  of  2  yrs.,  at  8^,  is  $348  ? 

Solution :  $1  will  amount  to  $1.16  in  2  yrs.  at 
8^> ;  therefore,  it  will  take  as  many  times  1  dollar 
to  amount  to  $348  as  1.16  is  contained  in  348,  or 
300  times  $1.  The  answer,  then,  is  $300.  What 
will  be  the  amount  of  this  principal  at  the  end  of 
1  yr.  ?  what  at  the  end  of  6  in.  ?  what  at  the  end 
of  4  yrs.  ? 

44.  What  is  the  principal  when  the  amount  at 
the  end  of  3  yrs.,  at  4^),  is  $448  ? 

What  would  this  principal  amount  to  at  the  end 
of  1  yr.  ?  at  the  end  of  2  yrs.  ? 

45.  What  is  the  principal  when  the  amount  at 
the  end  of  1  yr.  6  m.,  at  6^,  is  $218  ? 

What  would  this  principal  amount  to  at  the  end 
of  1  yr.  ?  at  the  end  of  2  yrs.  ?  at  the  end  of  3 
yrs.  3  d.  ? 

46.  What  is  the  principal  when  the  amount  at 
the  end  of  2  yrs.,  at  10^,  is  $60  ? 

What  would  this  principal  amount  to  at  the  end 
of  1  yr  ?  at  the  end  of  5  yrs.  ? 

47.  What  is  the  principal  when  the  amount  at 
the  end  of  2  yrs.  6  m.  18  d.,  at  6^,  is  $1000? 


A.]  Interest.  155 

Solution  : 

The  interest  on  $1  for  2  yrs.  is       .         .  $0.12 

The  interest  on  $1  for  6  m.  is         .         .     0.03 

The  interest  on  $1  for  1  m.  is  $0.005 

The  interest  on  $1  for  6  d.  (%  of  1  m.)  is  $0.001 

The  interest  on  $1  for  18  d.  (f  of  1  m.)  is    0.003 

The  interest  on  $1  for  2  yrs.  6  m.  18  d.  is  10.153 

The  amount  of  $1  for  2  yrs.  6  m.  18  d.  is  11.153 

If  $1  amounts  to  $1.153,  it  will  take  as  many 

times  11  to  amount  to  11000  as  $1.153  is  contained 

in  $1000. 

1.153)1000.000(867.302     Ans.  $867.302,  which 
9224  expressed    to     the 

7760  nearest      cent      is 

6918  $867.30 

8420 
8071 
3490 
3459 


3100 
2306 

48.  What  is  the  principal  when  the  amount  at 
the  end  of  4  yrs.  3  m.  12  d.,  at  6^,  is  $400  ?     Ans. 
(to  the  nearest  cent)  $318.22 

49.  What  is  the  principal  when  the  amount  at 
the  end  of  4  yrs.  2  m.,  at  6/0,  is  $180.     Ans.  $144. 

50.  What  is  the  principal  when  the  amount  at 
the  end  of  8  yrs.  2  m.,  at  6^,  is  $1024.     Ans.  (to 
the  nearest  cent)  $687.25 

51.  What  is  the  principal  when  the  amount  at 


156  Percentage.  [§  7. 

the  end  of  3  yrs.  4  m.,  at  6^,  is  13668  ?     Ans.  (to 
the  nearest  cent)  $3056.67 

Promissory      A  written  promise  to  pay  money  is  called 
Note-      a  PROMISSORY  NOTE,  or  simply  a  NOTE. 

52.  J.  G.  Bland  of  Augusta,  Maine,  who  bor- 
rowed from  A.  B.  Smith  on  Jan.  5,  1886,  $300, 
payable  in  3  years  from  date  (that  is  from  Jan. 
5,  1886),  with  interest  at  6^,  to  be  paid  annually, 
gave  Mr.  Smith  the  following  note  : 

$300.00  AUGUSTA,  ME.,  Jan.  5,  1886. 

For  Value  Received, 

I  promise  to  pay  A.  B.  Smith,  or  order,  three 
hundred  -j™-^  dollars  in  three  years  from  this  date, 
with  interest  at  6^,  to  be  paid  annually. 

J.  G.  BLAND. 

The  words  "  to  be  paid  annually  "  required  Mr. 
Bland  to  pay  at  the  end  of  each  year  (Jan.  5, 
1887,  Jan.  5,  1888,  and  Jan.  5,  1889)  the  interest 
then  due. 

How  much  interest  was  due  at  the  end  of  each 
year? 

The  words  "  or  order  "  after  Mr.  Smith's  name 
gave  him  the  power  to  order  Mr.  Bland  to  pay 
the  money  to  some  one  else.  Desiring  him  to  pay 
the  money  to  T.  F.  Brown,  he  would  write  on  the 
back  of  the  note,  — 

"  Pay  to  T.  F.  BROWN. 

A.  B.  SMITH." 

It  is  not  necessary  to  word  a  note  just  like  the 


A.]  Interest.  157 

foregoing ;  all  that  is  required  is  that  the  whole 
promise  shall  be  distinctly  stated.  It  is  advisable, 
though  not  absolutely  necessary,  that  the  person 
who  makes  the  promise  (the  maker  of  the  note,  as 
he  is  called)  should  acknowledge  by  some  such 
expression  as  "  For  Value  Received  "  that  he  has 
received  either  the  money  or  something  else  that 
he  considers  an  equivalent.  For  greater  distinct- 
ness the  sum  named  in  a  note  should  be  expressed 
both  in  words  and  by  figures  ;  this  sum 
is  called  the  FACE  of  the  note. 

53.  Write   a   note   that  you  would   give   your 
next-door  neighbor,  if   you  were  to  borrow  from 
him  $430.60,  payable  in  3  years  from  now,  with 
interest,  to  be  paid  annually,  at  6jfc. 

How  much  interest  would  you  have  to  pay  him 
at  the  end  of  each  year  ? 

54.  Write  a  note  for  each  one  of  the  following 
cases :  — 

A.  On  Feb.  1,  1885,  H.  M.  TwitcheU  bought  of 
J.  P.  True  a  tricycle,  for  which  he  promised  to 
pay  $150.50  on  March  16,  1885,  with  interest  at 
6/0. 

[Business  men  would  regard  the  time  here  as 
1  m.  15  d.,  or  1J  months  ;  national  governments, 
however,  would  count  the  exact  number  of  days 
(42),  and  would  consider  the  interest  to  be  ^g-  of 
the  interest  for  a  year.] 

B.  On  July  15,  1885,  Edward  Gilchrist  bor- 
rowed of  Frank  Allen  $450,  which  he  promised  to 
pay  on  Jan.  21,  1886,  with  interest  at  856.     [The 


158  Percentage.  [§  7. 

time  here  is  to  be  regarded  as  6  m.  6  d.,  or  6£ 
months.] 

C.  On  Dec.  24,  1885,  Willard  Small  bought  of 
Iloughton,  Mifflin  &  Co.  books  to   the  value  of 
1206.34,  which  he  promised  to  pay  on  Feb.  17, 
188G,  without  interest. 

D.  On  Feb.  1,  1885,  Clarke  &  Carruth  bought 
of  Roberts  Brothers  books  to  the  value  of  « 1000,  to 
be  paid  for  in  60  days,  with  interest  at  6jfc. 

[When  the  time  is  specified  in  days  the  exact 
number  of  days  must  be  counted  from  the  date  of 
the  note  to  find  when  the  money  is  due.  In  this 
case,  for  instance,  27  days  in  February,  plus  31 
days  in  March,  plus  2  days  in  April,  make  up  the 
60  days.  The  money  was  due,  then,  on  April  2.] 

E.  On  Feb.  15,  1885,  Richard  Roe  bought  of 
John  Doe  a  pair  of  horses  for  $600,  to  be  paid  for 
in  90  days,  with  interest  at  6^>. 

F.  On  July  31,  1885,  Amos  Ames  bought  of 
Byron  Burns  a  barn  for  $1500,  to  be  paid  for  in 
60  days,  with  interest  at  8^fc. 

G.  On  Sept.  1, 1888,  John  Jones  bought  of  J.  C. 
Ayer  11  acres  of  wood-land  for  §400,  to  be  paid 
for  in  90  days,  with  interest  at  5J^. 

55.  Compute  the  interest  in  each  of  the  preced- 
ing cases,  excepting  C. 

56.  This  morning  James   Pike  said  to  Irving 
Blake,  "  If  you  will  give  me  1106  for  my  bay  mare 
Dolly  you  need  not  pay  for  her  until  one  year 
from  date."     Mr.  Blake  accepted  the    offer,  and 
gave  Mr.  Pike  the  following  note  : 


A.]  Interest.  159 


*TTTO"  CAMBRIDGE,  MASS.,* 

For  Value  Received, 

I  promise  to  pay  James  Pike,  or  order,  one  hun- 
dred six  and  -f^  dollars  one  year  from  date,  with- 
out interest.  IRVING  BLAKE. 

A  few  minutes  after  the  note  was  given,  Mr. 
Pike  found  that  he  needed  some  money  at  once, 
and  therefore  offered  to  sell  the  note  to  M.  M. 
Sawin ;  now  supposing  that  Mr.  Sawin  has  money 
that  he  is  willing  to  invest  at  6^,  what  can  he 
afford  to  pay  for  the  note  ?  [Compare  with  Ex- 
ample 41.] 

Ans.  $100  ;  because  the  payment  of  $100  now 
will  insure  his  getting  $106  from  Mr.  Blake  a 
year  hence  ;  that  is,  in  one  year  he  will  get  back  his 
1100  and  in  addition  $6  (6^)  as  interest.  The 
$100  that  Mr.  Sawin  can  afford  to  pay  for  the 
note  now  is  called  the  PRESENT  WORTH  of  the 
note. 

The  PRESENT  WORTH  of  a  debt  due  at  some 
future  time,  without  interest,  is  the  sum    Present 
which,  put  at  interest,  will   amount  to     Wortb. 
the  debt  when  it  becomes  due. 

57.  Find  the  present  worth  on  Jan.  1,  1885, 
of  each  of  the  following  notes,  regarding  the  rate 
of  interest  in  getting  the  present  worth  of  A  to  be 
4fo  ;  of  B  to  be  8^ ;  of  C  to  be  4fi  ;  of  D  to  be 
6^;  and  of  E  to  be  Wjb. 

*  The  pupil  should  fill  in  the  date  of  the  day  on  which  he  per- 
forms this  example. 


160  Percentage.  [§  7. 

A. 

$108^°^.  SAN  FRANCISCO,  CAL.,  Jan.  1,  1885. 

For  Value  Received, 

I  promise  to  pay  B.  M.  Snow,  or  order,  one 
hundred  eight  and  y^  dollars  in  two  years  from 
this  date.*  A.  B.  COOK. 

[Compare  with  Example  42.] 

B. 

$348^.  NEW  YORK,  N.  Y.,  Jan.  1,  1885. 

For  Value  Received, 

I  promise  to  pay  D.  E.  Faunce,  or  order,  three 
hundred  forty-eight  and  -ffa  dollars  in  two  years 
from  this  date.  B.  C.  DOWD. 

[Compare  with  Example  43.] 

C. 

$448T^j.  CHICAGO,  ILL.,  Jan.  1,  1885. 

For  Value  Received, 

I  promise  to  pay  E.  F.  Griffin,  or  order,  four 
hundred  forty-eight  and  -^fa  dollars  in  three  years 
from  this  date.  C.  D.  EVARTS. 

[Compare  with  Example  44.] 

D. 

$218^.  NEW  ORLEANS,  LA.,  Jan.  1,  1885. 

For  Value  Received, 

I  promise  to  pay  F.  G.  Hale,  or  order,  two 
hundred  eighteen  and  -j^  dollars  in  one  year  and 
six  months  from  this  date.  D.  E.  FALES. 

[Compare  with  Example  45.] 

*  Where  nothing  is  said  ahoiit  interest,  no  interest  can  be 
claimed  if  the  sum  promised  is  paid  when  due. 


A.]  Interest.  161 

E. 

$^lTO*  INDIANAPOLIS,  IND.,  Jan.  1,  1885. 

For  Value  Received, 

I  promise  to  pay  G.  H.  Ives,  or  order,  sixty  and 
y^fo  dollars  in  two  years  from  this  date. 

E.  F.  GATES. 

[Compare  with  Example  46.] 

58.  Find  the  present  worth  on  July  1,  1885,  of 
each  of  the  notes  of  Example  57,  regarding  the 
rate  of  interest  in  each  case  the  same  as  before. 
[The  fact  that  July  1,  1885,  is  not  the  date  of 
each  note  has  nothing  to  do  with  the  question. 
All  that  we  are  required  to  find  is  the  sum  which, 
put  at  interest  on  July  1,  1885,  will  amount  in 
each  case  to  the  sum  promised  at  the  time  it  is 
due.] 

The  sum  which,  subtracted  from  the    True  Dis- 
face   of   a   note,  would   give  its  present    count- 
worth,  is  called  the  TRUE  DISCOUNT. 

The  True  Discount,  then,  is  the  interest  on  the 
present  worth. 

59.  Find  the  true  discount  on  Jan.  1,  1885,  on 
each  of  the  notes  of  Example  57. 

60.  Write    a   note    for   each  of   the   following 
cases,  and  find  the  true  discount  at  Gfi  of  each 
note  on  Jan.  1,  1885  : 

A.  On  Jan.  1,  1885,  A.  M.  Brown  bought  11500 
worth  of  groceries  of  C.  B.  Smith,  promising  to 
pay  for  them  in  6  months  from  date. 

B.  On  Jan.  1, 1885,  G.  H.  Irwin  bought  a  house 


162  Percentage.  [§  7. 

of  H.  I.  Knight  for  $2000,  promising  to  pay  for  it 
in  2  yrs.  6  m.  from  date. 

C.  On  Jan.  1,  1885, 1.  K.  Lyons  bought  a  horse 
of  K.  L.  Minot  for  $248,  promising  to  pay  for  him 
in  one  year  from  date. 

D.  On  Jan.   1,   1885,   L.   M.   Nason  bought 
$ 861.50  worth  of  boots  of  M.  N.  Ogdeu,  promising 
to  pay  for  them  in  6  months  from  date. 

61.  The  interest  on  the  face  of   the  note   of 
Example   60   A.   for   6   months,  at   6^,  is   $45, 
whereas  the  true  discount  is  $43.69,  therefore  the 
interest  on  the  face  of  the  note  exceeds  the  true 
discount  by  $1.31. 

Find  by  how  much  the  interest  on  the  face  of 
each  of  the  other  notes  of  Example  60  exceeds  the 
true  discount. 

62.  $1060.  OMAHA,  NEB.,  Jan.  16, 1886. 
For  Value  Received, 

I  promise  to  pay  James  Brown,  or  order,  one 
thousand  sixty  and  T^  dollars  in  one  year  from 
date.  J.  B.  SMITH. 

What  is  the  face  of  this  note  ? 

What  was  the  true  discount  on  Jan.  16, 1886,  at 
6^  ?  what  was  the  present  worth  ? 

[The  man  who  paid  $1000  (the  present  worth) 
for  this  note  on  Jan.  16,  1886,  is  said  to  have 
discounted  the  note  because  he  paid  for  it  the  face 
value  ($1060)  less  the  true  discount  ($60).] 

In  case  a  man  does  not  pay  his   debts  when 


A.]  Interest.  163 

they  become  due,  an  appeal  to  the  law  may  be 
made  to  make  him  pay  them.  The  maker  of  a 
note,  however,  —  excepting  a  note  payable  on  de- 
mand, —  need  not  pay  it  until  three  days  after  the 
time  of  payment  mentioned  in  the  note  :  that  is 
to  say,  a  note  is  not  legally  due  until  three  days 
after  it  is  nominally  due.  These  three  j)ays  Of 
additional  days  allowed  by  law  are  called  Grace. 

DAYS   OF   GRACE. 

The  maker  of  a  note,  payable  at  a  given  time 
with  interest,  may  be  compelled  legally  to  pay  in- 
terest for  the  three  days  of  grace ;  but  if  he  pays 
his  note  when  it  becomes  nominally  due  he  is  not 
usually  asked  to  pay  interest  for  the  days  of  grace, 
except  when  he  deals  with  a  bank. 

When  a  bank  discounts  a  note  it  deducts  a  dis- 
count larger  than  the  true  discount,  in  order  that 
it  may  receive  pay  for  its  trouble,  as  well  as  inter- 
est on  the  money  that  it  advances. 

The  BANK  DISCOUNT,  as  it  is  called,  on  a  note 
payable  at  a  specified  time,  without  in-       Bank 
terest,  is  the  interest  on  the  face  for  the       Discount, 
time  that  is  to  elapse  before  the  note  is  legally 
due.* 

63.  What  is  the  bank  discount  on  the  note  of 
Example  62  ?  Ans.  164.13.  [The  true  discount, 
as  we  have  seen,  is  only  f  60.] 

How  much  will  a  man  receive  who  gets  this  note 
discounted  at  a  bank  ?  Ans.  $995.87. 

*  As  has  been  stated,  it  is  not  legally  due  until  the  third  day 
uiter  the  time  of  payment  mentioned  in  the  note. 


164  Percentage.  [§  7. 

64.  How  much  would  each  of  the  following  notes 
have  brought  on  Jan.  1,  1886,  if  discounted  at  6^, 
bank  discount  ? 

Note  A,  for  $1632.12,  dated  July  1,  1885  ;  pay- 
able March  1,  1887. 

Note  B,  for  $56.28,  dated  Dec.  1,  1885 ;  pay- 
able June  1,  1886. 

Note  C,  for  $2361.18,  dated  Nov.  1, 1885 ;  pay- 
able Dec.  1,  1886. 

Note  D,  for  $1500,  dated  Jan.  1,  1886  ;  pay- 
able Sept.  1,  1886. 

65.  $100^.  CAMBRIDGE,  MASS.,  Sept.  1, 1887. 
For  Value  Received, 

I  promise  to  pay  the  Cambridge  National  Bank, 
or  order,  one  hundred  ^^  dollars  in  6  months 
from  date,  with  interest  at  6^>.  F.  D.  JONES. 

What  was  the  amount  of  this  note  at  the  time  it 
was  due  ?  A?is.  $103.05. 

What  is  the  face  of  a  note  of  the  same  date  and 
payable  at  the  same  time,  in  which  no  mention  is 
made  of  interest,*  that  will  give  the  bank  the 
same  amount  of  money  ?  Ans.  $103.05. 

66.  Make  a  note  equivalent  to  the  following,  of 
the  same  date  and  payable  at  the  same  time,  but 
without  interest : 

$566lTo-  BOSTON,  MASS.,  Sept.  1, 1887. 

For  Value  Received, 

I  promise  to  pay  the  Suffolk  National  Bank,  or 

*  It  will  be  remembered  that  where  no  mention  is  made  of 
interest,  none  can  be  collected  if  the  note  be  paid  when  due. 


A.]  Interest.  165 

order,   five   hundred    sixty-six  -f^   dollars   in    9 
months  from  date,  with  interest  at  4j6. 

67.  In  each  of  the  following  cases  find  the  face 
of  a  note,  payable  at  the  end  of  the  time  given,  but 
without  interest,  allowing  as  before  for  the  three 
days  of  grace : 

A.  $684.50,  due  in  1J  years,  at  5%. 

B.  $3984.00,  due  in  6  months,  at  6 ft. 

C.  $1000.00,  due  in  9  months,  at  4/o. 

68.  How  much  would  a  bank,  whose  rate  of  dis- 
count is  6^fc,  have  given  me  on  Sept.  1,  1887,  for 
a  note -for  $500,  dated  Sept.  1,  1887,  due  in  6 
months,  with  interest  at  4^>?     [Suggestion:  First 
find  the  face  of  an  equivalent  note  due  Sept.  1, 

1887,  without  interest,  and  then  compute  the  dis- 
count on  this  face.] 

69.  How  much  would  a  bank,  whose  rate  of  dis- 
count is  8/0,  have  given  me  on  Jan.  1,  1888,  for  a 
note  for  $1000,  dated  Jan.  1,  1888,  and  due  in  9 
months,  at  6yfo  ? 

70.  How  much  would  a  bank,  whose  rate  of  dis- 
count is  6^b,  have  given  me  on  Jan.  1,  1886,  for  a 
note  for  $750,  dated  Jan.  1,  1886,  and  due  in  18 
months,  at  4^fc  ? 

71.  How  much  would  a  bank,  whose  rate  of  dis- 
count is  6^fe,  have  given  me  on  June  1,  1888,  for  a 
note  for  $1500,  dated  June  1,  1888,  and  due  in  1 
year,  at  5^>  ?  how  much  would  it  have  given  me 
on  Jan.  1,  1888,  for  a  note  for  $800,  dated  Jan.  1, 

1888,  and  due  in  6  m.,  at  Sfi  ? 

72.  a.  How  much  would  a  bank,  which  discounts 


166  Percentage.  [§  7. 

at  6'/>,  have  given  me  on  Jan.  1,  1886,  for  a  note 
for  $1.00,  dated  Jan.  1, 1886,  and  payable,  without 
interest,  April  28,  1886  ?  Ans.  $0.98 

b.  How  large  a  note  should  I  have  had  to  give 
the  bank  in  order  to  get  $1.96  ?     [We  have  just 
seen  that  a  note  for  $1  would  have  brought  me 
$0.98;  therefore,  to  get  $1.96  my  note  would  have 
had  to  be  for  as  many  dollars  as  $0.98  is  contained 
in  $1.96,  that  is  for  $2.] 

c.  How  large  a  note  should  I  have  had  to  give 
a  bank  in  order  to  get  $2.94?    $3.82?    $9.80? 
$19.60?  $29.40?  $39.20?  $98.00?  $150?  $500? 
$1200? 

73.  If  I  want  a  bank,  whose  rate  of  discount 
is  6^f>,  to  pay  me  $979.50  to-day,  for  how  much 
must  I  make  my  note  payable  to  the  bank,  without 
interest,  in  4  m.  from  now  ?     [A  note  for  $1.00 
will  bring  now  $0.9795] 

74.  If  I  want  a  bank,  whose  rate  of  discount  is 
8^),  to  pay  me  $500  to-day,  for  how  much  must  I 
make  my  note  payable  to  the  bank,  without  inter- 
est, in  8  m.  from  now  ?     [A  note  for  $1.00  will 
bring  now  $0.946] 

75.  If  I  want  a  bank,  whose  rate  of  discount  is 
6^>,  to  pay  me  $2400  to-day,  for  how  much  must 
I  make  my  note  payable  to  the  bank,  without  in- 
terest, in  6  m.  from  now  ? 

76.  If  I  want  a  bank,  whose  rate  of  discount  is 
6^fc,  to  pay  me  $175  to-day,  for  how  much  must  I 
make  my  note  payable  to  the  bank,  without  inter- 
est, in  2  yrs.  from  now  ? 


B.]  Compound  Interest.  167 

Business  Customs  in  regard  to  Computing  Interest. 

A  month  is  regarded  as  ^  of  a  year.  (See 
foot-note  on  page  148.) 

In  computing  the  interest  for  a  given  number  of 
days  less  than  a  month,  a  day  is  regarded  as  ^  of 
a  month.  (See  foot-note  on  page  149.) 

In  computing  the  time  between  Jan.  1,  1885, 
and  July  20, 1885,  we  count  the  months  from  Jan. 
1  to  July  1,  and  then  the  days  from  July  1  to 
July  20,  and  thus  get  6  m.  19  d.  In  a  similar 
way  the  time  between  any  two  dates  is  computed. 

A  note  dated  Jan.  31,  and  payable  in  one  month, 
is  nominally  due  on  the  last  day  of  February,  that 
is,  on  February  28,  except  in  a  leap  year,  and  then 
on  Feb.  29  ;  and  a  note  dated  Feb.  28,  and  pay- 
able in  1  month,  is  due  March  28.  These  same 
principles  are  applied  in  other  similar  cases. 

When  a  note  falls  due  on  Sunday,  or  on  a  legal 
holiday,  it  is  payable  the  day  previous. 

B.    Compound  Interest. 

1.  a.  On  Jan.  1, 1880,  I  deposited  11000  in  the 
Bonanza  Savings  Bank.  The  rules  of  the  bank 
allow  depositors  4  </o  interest  payable  on  Jan.  1  of 
each  year,  on  such  sums  as  have  been  in  the  bank 
a  whole  year,  and  further  state  that  if  the  interest 
is  not  called  for  when  due  it  shall  be  immediately 
added  to  the  principal,  and  shall  itself  draw  interest 
just  as  if  it  were  a  new  deposit. 

I  drew  nothing  from  the  bank  until  Jan.  1, 1883, 


168  Percentage.  [§  7. 

but  then  called  for  all  the  money  standing  to  my 
credit ;  how  much  did  I  get,  and  how  much  interest 
had  my  11000  earned  ? 

Solution : 
$1000. 

.04 

$40.00  Interest  for  the  first  year. 
$1000.00 
$1040.00  Amount  to  my  credit  on  Jan.  1,  1881 

.04 

$41.60  Interest  for  the  second  year. 
$1040.00 

$1081.60  Amount  to  my  credit  on  Jan.  1,  1882. 
.04 

$43.264  Interest  for  the  third  year. 
$1081.60 

$1124.86  Amount  drawn  out  by  me  on  Jan.  1, 1883. 
$1000.00 
$124.86  Total  amount  of  interest. 

In  an  example  like  the  above  the  total  interest 
earned  is  called  COMPOUND  INTEREST. 

b.  What  would  my  $1000  have  earned  if  I  had 
been  allowed  only  the  ordinary  or  SIMPLE  INTER- 
EST, as  it  is  called? 

c.  If  the  Bonanza  Savings  Bank  had  allowed  its 
depositors  4  Jfc  a  year,  payable  semi-annually,  and 
had  further  agreed  to  add  immediately  to  the  prin- 
cipal interest  not  called  for  when  due,  how  much 
should  I  have  received  on  Jan.  1,  1883,  and  how 
much  would  the  interest  have  been  ? 


B.]  Compound  Interest.  169 

Solution : 
$1000. 

.02 

$20.00  Interest  for  the  first  half  year. 
$1000.       New  Principal  on  July  1, 1880. 
$1020. 

.02 

$20.40  Interest  for  the  second  half  year. 
$1020.00 

$1040.40  New  Principal  on  Jan.  1,  1881. 
.02 


).8080  Interest  for  the  third  half  year. 
$1040.40 
$1061.21  New  Principal  on  July  1,  1881. 

.02 

$21.2242  Interest  for  the  fourth  half  year. 
$1061.21 
$1082.43  New  Principal  on  Jan.  1,  1882. 

.02 

$21.6486  Interest  for  the  fifth  half  year. 
$1082.43 
$1104.08  New  Principal  on  July  1,  1882. 

.02 

$22.0816  Interest  for  the  sixth  half  year. 
$1104.08 

$112616  Total  on  Jan.  1,  1883. 
$1000.00 

$126.16  Total  amount  of  Interest. 
NOTE.     Interest  may  be  compounded  (added  to 


170  Perccntt/yc.  [§  7. 

the  principal)  once  a  year  or  once  each  half  year 
or  at  the  end  of  any  period  of  time  that  may  be 
agreed  upon ;  but  when  no  period  is  mentioned,  it 
is  understood  that  the  interest  (when  compounded) 
shall  be  compounded  annually. 

2.  At  compound  interest,  what  is  the  amount  of 
1200,  at  6jf>,  for  2  yrs.  6m.? 

$200. 
.06 

$12.00  Interest  for  the  first  year. 
$200. 

$212.00  New  Principal  at  the  end  of  first  year. 
.06 

$12.72  Interest  for  the  second  year. 
$212.00 
$224.72  New  Principal  at  the  end  of  second  year. 

.03 

$6.7416  Interest  for  6  months. 
$224.72 
$231.46  Amount  at  the  end  of  2  yrs.  6  m. 

3.  At   compound   interest,    compounded   semi- 
annually,  what  is  the  amount  of  $150,  at  4^>,  for 

1  yr.  9  m.  ? 

4.  At    compound   interest,   compounded   semi- 
annually,  what  is  the  amount  of  $3000,  at  6^fc,  for 

2  yrs.  2m.? 

5.  a.  At  compound  interest,  what  is  the  amount 
of  $1,  at  6jfc,  for  1  yr.  ?  for  2  yrs.  ?  for  3  yrs.,  etc., 
up  to  8  yrs.  ? 


B.] 


Compound  Interest. 


171 


Arrange  the  results,  carried  out  to  six  places  of 
decimals,  in  a  table,  as  indicated  below. 

Years.  Amount. 

1  1.060000 

2 

3 

4 

5 

6 

7 

8 

b.  Knowing  the  amount  of  $1  for  5  years  at  6^ 
compound  interest,  how  can  we  find  the  amount  of 
$3  for  the  same  time  ? 

c.  What  is  the  amount  of  17000  for  8  years? 
[Refer  to  the  table  that  you  have  just  formed  for 
the  amount  of  $1  for  8  years.] 

6.  At  compound  interest,  what  is  the  amount  of 
$200,  at  6^,  for  2  yrs.  6m.?     [Take  the  amount 
of  $1  for  2  yrs.  from  the  table  of  the  preceding 
example,  and  from  this  find  the  amount  of  $200 
for  the  same  time,  then  compute  and  add  in  the 
interest  for  6  m.    Compare  the  result  with  that  ob- 
tained in  Ex.  2.] 

7.  At  compound  interest,  what  is  the  amount  of 
$1800  at  6^b  for  5  yrs.  8m.? 

8.  At  compound  interest,  what  is  the  amount  of 
$2500  at  6/0  for  7  yrs.  4  m.  ? 

NOTE.  A  table  showing  the  amount  of  $1  at 
compound  interest  from  1  year  to  50  years,  inclu- 
sive, at  3,  3£,  4,  4j,  5,  6,  and  7  per  cent  is  given 
on  page  282. 


172  Percentage.  [§  7. 

(7.    Partial  Payments. 

1.  I  owed  $100  at  6^  :  At  the  end  of  8  months 
I  paid  $54;  how  much  did  I  then  owe?  At  the  end 
of  18  months  I  paid  in  full ;  how  much  did  I  then 
have  to  pay  ? 

Solution : 

$100.00 

Interest  on  $100  for  8  m.  .         .         $4.00 

Amount  at  the  end  of  8  m.    .         .         .     $104.00 
Paid  at  the  end  of  8  m.          .         .         .       $54.00 

Unpaid  at  end  of  8  m $50.00 

Interest  on  $50  for  10  m.                .         .         $2.50 
Due  at  end  of  18  m $52.50 

2.  I  owed  $1000  at  6J& :  At  the  end  of  6  m. 
I  paid  $400,  and  at  the  end  of  18  m.  $300.    How 
much  did  I  owe  at  the  end  of  18  m.  ? 

Ans.  $367.80 

3.  $500T^.  BOSTON,  MASS.,  July  1,  1885. 
For  Value  Received, 

I  promise  to  pay  A.  B.  Coes,  or  order,  five  hun- 
dred —fa  dollars  on  demand,  with  interest  at  6jfc. 

B.  C.  DOLE. 

Payments  $50  Jan.  1,  1886;  $150  Sept.  1, 
1886  ;  $100  Nov.  1,  1886.  How  much  was  due 
Jan.  1,1887? 


4.  $1000T^-.  ALBANY,  N.  Y.,  Jan.  1,  1884. 

For  Value  Received, 

I  promise  to  pay  C.  D.  Earle  one  thousand  -ffi 
dollars  on  demand  with  interest  at  4/o. 


C.]  Partial  Payments.  173 

Payments  $400  Dec.  1,  1884  ;  $200  March  1, 
1885  ;  $300  Jan.  1,  1886.  How  much  was  due 
July  1,  1886  ? 

5.  I  owed  $10000  at  6fo.     At  the  end  of  4  m. 
I  paid  $300  ;  at  the  end  of  7  m.  $100  ;  and  at  the 
end  of  12  m.  $800.     How  much  did  I  owe  at  the 
end  of  16  m.  ? 

$10000.00 

Interest  on  $10000  for  4m.        .         .         $200.00 
Amount  of  $10000  at  end  of  4  m.  .         $10200.00 

Paid  at  end  of  4  m $300.00 

Unpaid  at  end  of  4  m.        .         .         .       $9900.00 

Interest  on  $9900  for  3  m.  $148.50 
Amount  of  $9900  for  3  m.  $10048.50 
Paid  at  end  of  7  m.  $100.00 

Here  the  payment  ($100)  is  less  than  the  interest  ($148.50) 
earned  since  the  last  payment.  The  Supreme  Court  of  the  United 
States  has  decided  that  such  a  payment  must  not  be  subtracted 
from  the  amount,  but  must  be  added  to  the  next  payment,  and  be 
treated  as  if  paid  at  the  same  time  with  the  next  payment.  In 
this  case,  then,  we  add  the  $100  to  the  next  payment  ($800),  and 
proceed  as  if  $900  were  paid  at  the  end  of  12  m.  and  nothing  at 
the  end  of  7  m. 

Unpaid  at  end  of  4  m.          .         .         .  $9900.00 

Interest  on  $9900  for  8  m.  .         .         .  $396.00 

Amount  of  $9900  for  8  m.  .         .         .  $10296.00 

Paid  at  end  of  12  m $900.00 

Unpaid  at  end  of  12  m.        .         .         .  $9396.00 

Interest  on  $9396.00  for  4  m.       .         .  $187.92 

Due  at  end  of  16  m $9583.92 

6.  I  owed  $10000  at  6fi.     At  the  end  of  4  m. 
I  paid  $800,  at  the  end  of  8  m.  $100,  at  the  end  of 


174  Percentage.  [§  7. 

12  ra.  $100,  and  at  the  end  of  16  m.  $1000 ;  how 
much  did  I  owe  at  the  end  of  18  m.  ? 

[If,  after  adding  a  payment  to  the  next,  as  in  the  last  example, 
we  find  that  the  sum  of  the  two  payments  is  less  than  the  interest 
earned  at  the  time  of  this  next  payment,  we  add  this  sum  to  the 
following  payment,  and  so  on  until  we  get  a  sum  that  equals  or 
exceeds  the  interest.] 

Ans.  $8851.64 

7.  $346.36  CAMBRIDGE,  MASS.,  March  26,  1880. 
For  Value  Received, 

I  promise  to  pay  A.  B.  Clark,  or  order,  three 
hundred  forty-six  and  -j3^  dollars  on  demand,  with 
interest  at  QJb.  B.  C.  DOLE. 

Payments  July  20,  1880,  $54.75 ;  April  8, 1881, 
$10  ;  Sept.  26, 1881,  $5.50;  Jan.  6,  1882,  $150.46. 
What  was  due  May  2,  1882  ?  Ans.  $161.43 

NOTE.  The  method  just  indicated  of  treating 
part  or  PARTIAL  PAYMENTS  —  called  the  United 
States  method  —  has  been  adopted  by  most  states. 
Where,  however,  a  settlement  is  made  in  a  year  or 
less,  the  Merchants'  method,  indicated  in  the  next 
example,  is  often  used. 

8.  A  owed  B  $100,  payable  in  1   yr.    at  6^> : 
at  the  end  of  4  m.  A  paid  B  $50  ;  how  much  did 
he  owe  him  at  the  end  of  the  year? 

By  the  Merchants'  method  a  payment  is  regarded 
as  a  loan,  to  be  accounted  for  at  the  time  of  settle- 
ment. In  this  example,  then,  at  the  time  of  settle- 
ment we  regard  B  as  having  loaned  A  $100  for  1 
yr.  and  A  as  having  loaned  B  $50  for  8m.;  there- 
fore, at  the  time  of  settlement,  A  owes  B  the 


C.]  Partial  Payments.  175 

amount  of  $100  for  1  yr.,  or  $106.00 

and  B  owes  A  the  amount  of  $50  for 

8  m.,  or  $52.00 

Balance  to  be  paid  by  A  to  B,  $54.00 

9.  On  July  1,  1884,  Amos  Brown  bought  of 
Byron  Coe  $1000  worth  of  boots,  for  which  he 
gave  his  note  at  6^),  payable  in  1  year.  Brown 
made  payments  as  follows  :  Sept.  1,  $200  ;  Dec.  1, 
$300;  Feb.  1,  1885,  $100.  How  much  did  he 
owe  Mr.  Coe  on  July  1,  1885  ? 

By  the  Merchants'  method  no  balance  is  found 
until  the  time  of  settlement,  at  which  time  we  say 
that  on  the  one  hand  Brown  owed  Coe  the  amount 
of  $1000  for  1  yr.,  or  $1060.00 

and  on  the  other  hand  Coe  owed  Brown 
the  amount  of  $200  for  10  m.,  or  $210.00 
$300  for  7  m.,  or    $310.50 

$100  for  5  m.,  or    $102.50 

Total  $623.00 

Balance  owed  by  Brown  on  July  1,  1885,  $437.00 

1O.  $387.75  BOSTON,  MASS.,  May  15,  1880. 

For  Value  Received, 

I  promise  to  pay  Charles  Doe  three  hundred 
eighty-seven  -£$$  dollars  on  demand  with  interest 
at  6/0.  DAVID  EMERY. 

Payments  July  21,  1880,  $75  ;  Oct.  10,  1880, 
$125  ;  Feb.  24,  1881,  $50.  The  account  was  set- 
tled on  May  15,  1881,  what  was  then  due  ? 

Ans.  $152.19 


176  Percentage.  [§  7. 

11.  On  a  6%  note  of  $1263  dated  Jan.  1,  1886, 
the  following  payments  were  made  : 

Feb.  1,  1886,  1100  ;  March  16,  1886,  $150 ; 
April  30,  1886,  $200. 

How  much  was  due  July  1,  1886  ? 


D.  Equation  of  Payments. 

1.  In  how  long   a  time  will    1    dollar   gain  as 
much  interest  as  $15  will  gain  in  a  month  ? 

2.  In   how  long  a  time  will  1    dollar   gain   as 
much  interest  as  $8  will  gain  in  3  months  ? 

3.  In  how  long  a  time  will  1  dollar  gain  as 
much  interest  as  $24  will  gain  in  5  months  ? 

4.  In  how  long  a   time  will   1  dollar  gain  as 
much  interest  as  $158  will  gain  in  11  months? 

5.  In  how  long  a  time  will  $3  gain  as   much 
interest  as  1  dollar  will  gain  in  24  months  ? 

6.  In  how  long  a  time  will  $28  gain  as  much 
interest  as  1  dollar  will  gain  in  157  months  ? 

7.  A  lent  B  $8,  which  was  paid  in  2  months ; 
afterwards  B  lent  A  1  dollar.     How  long  should 
A  keep  the  1  dollar  in  order  to  compensate  him- 
self for  his  loan  to  B  ? 

8.  C  lent  D   1  dollar,   which    was   paid  in  15 
months  ;    afterwards   D   lent   C    $5.      How   long 
should  C  keep  the  $5  in  order  to  compensate  him- 
self for  his  loan  to  D  ? 

9.  A  borrowed  of  B  $17  for  11  months,  prom- 
ising him  a  like  favor.    Afterwards  B  lent  A  $25 ; 
how  long  ought  he  to  keep  it  to  balance  the  favor  ? 


D.]  Equation  of  Payments.  177 

Suggestion:  Find  how  long  he  should  keep  1 
dollar,  and  then  from  this  how  long  he  should 
keep  25  dollars. 

10.  I  lent  a  friend  $257,   which  he  kept  15 
months,  promising  to  do  me  a  like  favor,  but  he 
was  not  able  to  loan  me  more  than  $100  ;    how 
long  could  I  keep  it  ? 

11.  A  owes  B  notes  to  be  paid  as  follows  :  $7 
to  be  paid  in  3  months,  and  $5  to  be  paid  in  8 
months  ;  but  he  wishes  to  pay  the  whole  at  once. 
In  what  time  should  he  pay  it  ? 

Solution :  $7  for  3  months  is  the  same  as  1  dollar 
for  21  months  ;  and  $5  for  8  months  is  the  same 
as  1  dollar  for  40  months.  He  may  have  1  dollar 
40  +  21,  or  61  months;  the  question  now  is  how 
long  may  he  keep  7  +  5,  or  12  dollars.  It  is  evi- 
dent he  may  keep  it  -^  of  61  months,  or  5  months 
and  2  days. 

12.  C   owes   D   $380,  to  be  paid  as  follows: 
$100  in  6  months  ;  $120  in  7  months ;  and  $160  in 
10  months.     He  wishes  to  pay  the  whole  at  once. 
In  how  many  months  is  it  due  ? 

13.  A  merchant  has  due  him  300X,  to  be  paid 
as  follows  :  50<£  in  2  months  ;  100<£  in  5  months ; 
and  the  rest  in  8  months.     It  is  agreed  to  make 
one  payment  of  the  whole.     In  how  many  months 
is  it  due  ? 

14.  F  owes  H  $1000,  of   which  $200  is   now 
due,  $400  is  due  in  5  months,  and  the  rest  in  15 
months.     F  wishes  to  make  one  payment  of  the 
whole.     Required  the  time  ? 


178  Percentage.  [§  7. 

15.  A  merchant  has  due  him  a  certain  sum  of 
money,  of  which  £  is  to  be  paid  in  2  months,  -J-  in 
3  months,  and  the  rest  in  6  months.    In  what  time 
is  the  whole  due  ? 

16.  a.  I  owe  1200  to  be  paid  in    3   months, 
'$500  to  be  paid  in  4  months,  and  $100  to  be  paid 
in  6  months.    In  how  many  months  must  I  make 
one  payment  of  the  whole  in  order  to  cancel  my 
indebtedness  ?     Arts.  In  4  months. 

6.  How  much  should  I  have  to  pay  now  in  order 
to  cancel  my  indebtedness  if  6<fc  is  the  rate  of  in- 
terest? 

[Find  the  present  worth  of  $800  due  in  4 
months.] 

17.  I  owe  $100  due  in  12  months,  $400  due  in 
10  months,  $300  due  in  8  months,  and  $200  due 
in  6  months.    How  much  must  I  pay  now  in  order 
to  cancel  my  debts  if  6^6  is  the  rate  of  interest  ? 

18.  I  owe  $200  that  should  have  been  paid  3 
months  ago,  $500  due  in  4  months,  and  $300  due 
in  6  months.     When  should  I  make  one  payment 
of  the  whole  in  order  to  cancel  the  debts  ? 

Solution  :  $500  for  4  m.  =  $1  for  2000  m. 

$300  for  6  m.  =  $l  for  1800  m. 

$500  for  4  m.  and  $300  for  6  m.  =  $l  for  3800  m. 
I  have  had  $200  for  3  m.  =  $lfor    600m. 

I  can  keep  $1  for  3200  m. 

or  $1000  for  f$£g-  m. 

fffa  =  3.2  m.  =  3  m.  12  d.  Ans. 

19.  A  merchant   has  three  notes  due   him  as 


E.]  Stocks.  179 

follows  :  one  of  $300  due  in  2  months ;  one  of 
$250  due  in  5  months ;  and  one  of  $180  due  3 
months  ago ;  the  whole  of  which  he  wishes  to  re- 
ceive now.  What  ought  he  to  receive,  allowing  6 
per  cent  interest  ? 

[The  present  worth  of  $730  due  in  1.8  months 
nearly,  or  1  month  and  24  days  =  $723.43  Ans. 

20.  A   gave  B  four  notes   as  follows :   one  of 
$75,  dated  June  5,  1879,  to  be  paid  in  4  months  ; 
one  of  $150,  dated  August  15,  to  be  paid  in  6 
months ;  one  of  $170,  dated  September  11,  to  be 
paid  in  5  months  ;  and  one  of  $300,  dated  Novem- 
ber 15,  to  be  paid  in  3  months.     They  were  all 
without  interest  until  they  were  due.     On  January 
1,  1880,  he   proposed   to   pay  the   whole.    What 
should  he  have  paid  ? 

21.  A  owes  B  $158.33,  due  in  11  months  and 
17  days,  which  he  proposes  to  pay  at  once.     What 
ought  he  to  pay,  if  the  rate  of  interest  is  5  per 
cent? 


E.  Stocks. 

In  the  year  1864  the  inhabitants  of  the  flourish- 
ing town  of  Draco  were  anxious  to  have  a  horse 
railroad  from  a  house  called  the  Old  Howe  Tavern 
to  a  large  shoe  factory  on  the  opposite  side  of  the 
town.  Some  of  the  richest  men  of  the  town  got 
together  and  offered  to  subscribe  $100,000  toward 
building  the  road,  with  the  understanding  that  each 
subscriber  should  receive  from  the  yearly  earnings 


180  Percentage.  [§  7. 

a  sum  proportional  to  the  amount  of  his  subscrip- 
tion. An  association  was  formed  which  petitioned 
the  state  legislature  to  be  incorporated  as  the 
Draco  Horse  Railroad  Company,  with  the  privi- 
leges necessary  to  build  and  manage  the  road ;  the 
petition  was  granted.  The  money  that  had  been 
subscribed  was  collected  and  formed  what  was 
called  the  CAPITAL  STOCK  of  the  company.  The 
stock  was  divided  into  shares  of  f  100  each,  and 
a  certificate  was  given  to  each  subscriber  by  the 
officers  of  the  company,  stating  how  many  shares 
belonged  to  him,  and  that  each  share  represented 
$  100.  The  subscribers  thus  became  what  are  called 

STOCKHOLDERS  Or  SHAREHOLDERS. 

During  the  first  year  the  road  was  built,  but 
earned  no  money ;  during  the  second  year  the 
road  earned,  above  all  expenses,  $5,000,  which 
was  divided  among  the  stockholders,  in  sums  pro- 
portional to  the  number  of  shares  of  stock  owned 
by  each ;  these  payments  were  called  DIVIDENDS. 

1.  How  large  a  dividend  did  a  person  receive 
who  owned  only  one  share  of  stock  ?  what  per  cent 
a  year  had  his  money  earned  since  he  first  paid 
it  in? 

During  the  third  year  the  burning  of  the  factory 
caused  travel  to  decrease  to  such  an  extent  that 
the  road  earned  only  enough  to  pay  its  expenses. 
Many  of  the  stockholders  losing  confidence  in  the 
road  wanted  to  sell  their  shares  of  stock,  but  no 
one  was  willing  to  buy  them  at  the  par  value  (§100 
each)  ;  several  shares  were  sold  at  $88  a  share,  or 


E.]  Stocks.  181 

as  it  is  more  briefly  expressed  were  sold  at  88. 
Opinions  were  divided  as  to  the  success  of  the  road 
in  the  future  ;  some  people  were  anxious,  there- 
fore, to  sell  stock  and  others  to  buy  it ;  a  STOCK 
BROKER  (a  person  who  makes  a  business  of  buying 
and  selling  stocks)  was  generally  employed  to  buy 
for  those  who  wanted  to  buy,  and  to  sell  for  those 
who  wanted  to  sell ;  his  charge  or  COMMISSION  for 
buying  or  selling  was  i  of  VJo  of  the  par  value  of 
the  shares  that  he  bought  or  sold. 

2.  How  much  was  his  commission  for  selling  15 
shares  ?  how  much  for  buying  28  shares  ? 

Some  speculators,  wishing  to  buy  the  stock  as 
low  as  possible,  started  rumors  that  the  factory 
would  not  be  rebuilt  for  several  years,  and  that 
the  population  of  the  town  would  never  be  as  great 
as  heretofore,  and  tried  in  every  way  possible  to 
make  the  stockholders  believe  that  the  road  would 
be  a  failure,  and  thus  to  make  them  willing  to  sell 
their  shares  at  a  low  rate.  Such  persons,  who  try 
to  bear  down  or  depress  the  prices  of  stocks,  are 
called  the  BEARS  of  the  stock  market.  These  same 
speculators,  having  bought  up  a  large  number  of 
shares  at  a  low  price,  waited  until  the  factory  was 
rebuilt ;  they  then  started  rumors  as  to  the  proba- 
ble great  prosperity  of  the  road,  and  afterward  suc- 
ceeded in  selling  their  stock  for  a  good  deal  more 
than  they  gave  for  it ;  in  trying  to  make  the  prices 
of  stock  high  (to  toss  them  up  as  it  were)  they 
became  what  are  called  the  BULLS  of  the  stock 
market. 


182  Percvntuyc.  [§  7. 

3.  a.  Mr.  Arthur  Brown  bought  10  shares  at  par 
($100  a  share)  ;  during  the  four  years  following  he 
received  only  one  5fi  dividend ;    he  then  sold  his 
shares  at  79.     In  both  buying  and  selling  he  em- 
ployed a  stock  broker,  who  charged  a  commission 
of  J  of  lj&  on  the  par  value.     How  much  money 
did  Mr.  Brown  lose?  what  per  cent  of  his  invest- 
ment ? 

b.  How  much  more  money  would  Mr.  Brown 
have  had  at  the  end  of  the  four  years,  if,  instead 
of  buying  stock  with  his  $  1,000,*  he  had  loaned  it 
to  the  Bonanza  Savings  Bank  at  4jfc  compound 
interest  (interest  compounded  annually)  ? 

4.  The  man  who  bought  Mr.  Brown's  ten  shares 
at  79  kept  them  three  years,  receiving  at  the  end 
of  the  first  year  a  dividend  of  2$fc ;  at  the  end  of 
the  second  year  a  dividend  of  3^,  and  at  the  end 
of  the  third  year  a  dividend  of  4$fc.     He  then  sold 
at  82.     He  did  not  employ  a  broker  either  in  buy- 
ing or  selling.     How  much  money  did  he  make  ? 
what  per  cent  a  year  on  his  investment  ?     Did  he 
make  more  or  less  than  if  he  had  loaned  his  money 
to  the  Bonanza  Savings  Bank,  and  how  much  ? 

5.  Mr.  A.  B.  Cranch  bought  15  shares  at  82, 
which  he  kept  for  four  years,  receiving  a  4^  divi- 
dend at  the  end  of  each  year.     He  then  sold  at  82. 
What  per  cent  a  year  did  he  make  on  his  invest- 
ment? 

6.  Mr.  B.  C.  Dowd  bought  45  shares  at  82,  which 
he  kept  for  15  years ;  during  the  first  5  years  he 
received  a  yearly  dividend  of  6^  ;  during  the  sec- 

*  Plus  the  broker's  commission  of  $2.50,  making  the  total  $1002.50. 


E.]  Stocks.  183 

ond  5  years,  a  yearly  dividend  of  8yfe ;  and  during 
the  third  5  years  a  yearly  dividend  of  10^.  He 
then  sold  at  132.  How  much  money  did  he  make? 
what  per  cent  a  year  on  his  investment  ? 

7.  Mr.  C.  D.  Evarts  bought  15  shares  of  East- 
ern Railroad  stock  at  52,   and  after  waiting  5| 
years  without  receiving  any  dividends,  he  sold  at 
118.     What  per  cent  a  year  did  he  make  on  his 
investment  after  deducting  J  of  Ijfc  to  a  broker  for 
buying  and  the  same  for  selling? 

8.  Mr.  D.  E.  Faunce  bought  18  shares  of  the 
Union  Pacific  Railroad  stock  at  88,  and  two  years 
later,  without  having  received  any  dividends,  was 
obliged  to  sell  at  48.     He  employed  a  broker,  both 
in  buying  and  selling ;  how  much  did  he  lose  ? 

9.  a.  I  have  $1000  which  is  earning  6$fe  a  year. 
How  much  shall  I  gain  or  lose  each  year  by  invest- 
ing this  money  in  the  stock  of  the  Chicago,  Bur- 
lington, and  Quincy  Railroad  at  f  134  a  share,  if 
each  share  or  portion  of  a  share  pays  an  annual 
dividend  of  8*/o  ? 

b.  In  order  to  get  6^>  a  year  on  my  money, 
how  much  per  share  can  I  afford  to  pay  for  stock 
which  pays  a  dividend  of  8fi  a  year  ? 

10.  I  bought  some  mining  stock  at  182,  which 
pays  a  dividend  of  10^)  a  year.  What  per  cent  a 
year  do  I  get  on  my  investment? 

NOTE.  Below  is  given  a  partial  list  of  the  sales 
of  stock  at  Boston  on  April  24,  1886.  The  figures 
on  the  left  indicate  the  number  of  shares  and  those 
on  the  right  the  number  of  dollars  received  for 
each  share. 


184 


Percentage, 


[§7. 


11.  Find  how  much  was  received  for  each  lot  of 
shares  and  the  brokers'  commission  on  each  lot. 


Boston  Stock 

Exchange.  —  April  24. 

BAILKOADS. 

100  Atch.,  T.,  &  St.  Fe, 

87 

70  Union  Pacific, 

50J 

325  Union  Pacific. 

511 

150             u 

50J 

181 

61* 

470            " 

51 

215 

51$ 

325 

51£ 

100 

51* 

25 

51i 

260 

61J 

15  Atch.,  T.,  &  St.  Fe*, 

86* 

219 

61* 

1140 

86} 

430 

52 

110                 " 

B6{ 

100 

52 

3 

86 

108  N.  Y.  &  N.  E., 

37J 

5  Fl.  &  P.  Mar., 

96^ 

100 

37* 

4  Chi.,  Bur.,  &  Q., 

134 

550 

374 

85 

134 

110  Clev.  &  Can., 

22 

50  M.,  II.,  &  0., 

34i 

100  Cin.,  S.,  &  Clev., 

15 

15  Mex.  Central, 

7} 

7  Bost.  &  Lowell, 

127J 

MINING  COMPANIES. 

60 

127 

19  Cal.  &  Hecla, 

227* 

5  Bost.  &  Albany, 

189 

10 

227 

2  Eastern, 

so* 

100  Bost.  &  Montana, 

oi 

3  Old  Colony, 

165* 

5 

165* 

LAND   COMPANIES. 

47  Metropolitan  H., 

80± 

400  Bost.  W.  Power, 

81 

5  Chi.,  Bur.,  &  Q., 

134 

200              " 

8A 

148             " 

133g 

25  Brookline, 

3f 

400  N.  Y.  &  N.  E., 

37 

10  Cambridge  H., 

83* 

MISCELLANEOUS. 

300  Atch.,  T.,  &  St.  F<$, 

86J 

3  Am.  Bell  Tel., 

162* 

3 

87 

50  N.  E.  Teleph., 

32? 

350                " 

86J 

50 

33 

1000                " 

87 

15  Am.  Bell  Tel., 

162J 

NOTE.  A  large  number  of  the  best  examples  of 
stocks  can  be  found  in  any  daily  paper  that  makes 
a  specialty  of  reporting  the  stock  exchange. 


F.]  Taxes.  185 

12.  Define  the  terms  Capital  Stock  ;  Stockhold- 
ers or  Shareholders  ;    Dividend ;    Stock   Broker ; 
Commission  ;  Bears  ;  Bulls. 

13.  What  is  meant  by  the  expression  "Union 
Pacific  Eailroad  Stock  sells  at  50j  "  ? 


F.    Taxes. 

A  town  needs  money  for  various  purposes,  such 
as  — 

Building  and  repairing  school-houses,  and  paying 
the  salaries  of  teachers. 

Making  and  repairing  roads  and  bridges. 

Supporting  a  fire  department. 

Supporting  a  police  department. 

Building  and  repairing  public  buildings  —  a 
town-house,  a  library  building,  etc. 

A  town  must  also  pay  its  share  of  the  expenses 
of  the  county  and  state  in  which  it  is  situated. 

Money  needed  for  the  purposes  just  mentioned 
and  for  other  similar  purposes  is  obtained  from  the 
inhabitants,  each  of  whom  is  required  to  pay  what  is 
considered  to  be  his  share  of  the  money  to  be  spent 
for  the  good  of  all. 

Money  obtained  in  this  way  is  called  a  TAX* 
Each  person's  share  is  determined,  in  most  states, 
somewhat  as  follows : 

First.  Every  male  citizen  who  has  reached  the 
age  of  21  years  is  required  to  pay  what  is  called  a 
poll  tax.  In  Massachusetts  this  tax  is  12.00,  one 


186  Percentage.  [§  7. 

half  of  which  is  paid  to  the  county  treasurer  and 
the  other  half  to  the  state  treasurer. 

Second.  Each  property  owner  is  required  to  pay 
an  amount  proportional  to  the  value  of  what  he 
owns.* 

1.  In  the  town  of  X,  Mass.,  the  amount  of  the 
entire  tax  to  be  raised  is  $  10,000 ;  the  number  of 
persons  who  must  pay  a  poll  tax  ($2.00)  is  500 ; 
the  value  of  all  the  property  owned  in  the  town  is 
$600,000.     How  large  a  tax  must  be  paid  by  a 
male  citizen  over  21  years  old,  the  value  of  whose 
taxable  property  is  $12,000  ? 

[The  amount  of  all  the  poll  taxes  is  500  x  $2,  or 
$1000.  There  remain,  then,  $9000  to  be  raised 
from  property  valued  at  $600,000  ;  therefore,  from 
property  valued  at  $1  there  would  be  raised  ^ow8l7 
dollars,  or  $0.015.  The  rate  of  taxation  in  this 
case  may  be  expressed  as  1JJ6,  or  15  mills  on  a 
dollar,  or  $15  on  a  thousand  dollars.] 

2.  The  citizens  of  the  town  of  Draco  have  voted 
to  raise  a  tax  of  $17,400 ;  the  number  of  inhabit- 
ants who  are  subject  to  a  poll  tax  of  $2  apiece  is 
450 ;   the  real  estate  f  owned  in  the  town  is  valued 
at  $850,000,  and  the  personal  property  at  $250,000. 
What  is  the  total  tax  of  B,  who  pays  one  poll  tax, 
and  whose  property  is  valued  at  $15,800  ?     What 
is  the  rate  of  taxation  ? 

*  Certain  kinds  of  property,  such  as  government  bonds,  etc., 
are  exempt  from  taxation. 

t  Immovable  property,  such  as  land  and  building's,  is  called 
real  estate ;  movable  property,  such  as  horses,  cows,  money,  etc., 
is  called  personal  property. 


F.]  Taxes.  187 

3.  The  tax  to  be  raised  in  a  town  is  $  10,600. 
The  property  is  valued  at  11,250,000,  and  there 
are  300  inhabitants  who  are  subject  to  a  $2  poll 
tax. 

What  is  the  total  tax  of  a  man  who  has  to  pay 
one  poll  tax  and  whose  property  is  valued  at 
$7,500  ?  What  is  the  rate  of  taxation  ? 

4.  In  a  town  where  the  rate  of  taxation  is  11 
mills  on  a  dollar,  what  is  the  total  tax  of  a  man 
who  pays  one  $2  poll  tax,  and  whose  property  is 
valued  at  112,800? 

5.  In  a  town  where  the  property  is  valued  at 
$3,300,000  the  total  tax  to  be  raised  is  $41,400  ; 
the  number  of  polls  at  $2  each  is  900. 

What  is  the  total  tax  of  Mr.  Brown,  who  pays 
one  poll  tax,  and  whose  property  is  valued  at 
$24,000  ?  What  is  the  rate  of  taxation  ? 

How  much  more  would  Mr.  Brown's  tax  have 
been  if  it  had  been  voted  to  raise  $5,800  more  for 
a  new  school-house  ?  What  would  the  rate  of  tax- 
ation have  been  in  this  case  ? 

6.  A's  entire  tax  is  $306  ;  he  pays  one  poll  tax 
of  $2,  and  the  rate  of  taxation  is  8  mills  on  a  dol- 
lar.    What  is  the  value  of  his  property  ? 

7.  What  is  the  value  of  the  property  in  a  town 
where  a  tax  of  $30,937.50  is  to  be  raised,  at  a  rate 
of  7^  mills  on  a  dollar? 

8.  The  richest  man  in  a  city  where  the  rate  of 
taxation  is  15  mills  on  a  dollar  pays,  in  addition 
to  his  poll  tax  of  $2,  a  tax  of  $22,500.     What  is 
the  value  of  his  property,  and  how  much  could  he 


188  Percentftye.  [§  7. 

save  each  year  in  taxes  by  moving  to  a  town  where 
the  rate  of  taxation  is  only  12  mills  on  a  dollar? 

9.  If  the  property  of  a  city  be  valued  at  $250,- 
000,000,  and  a  property  tax  of  14,000,000  is  to  be 
raised,  what  tax  (including  a  $2  poll  tax)  must  a 
man  pay  whose  property  is  valued  at  $15,000? 

10.  How  much  will  a  tax  collector  get  for  col- 
lecting a  tax  of  $20,000  if  he  is  paid  a  commission 
of  2^>  on  what  he  collects  ? 

11.  How  much  will  a  tax  collector  get  for  col- 
lecting a  tax  of  $95,000  if  he  is  paid  a  commission 
of  l£Jf>  on  what  he  collects  ? 

12.  If  a  tax  collector  whose  commission  is  lg^> 
receives  $640  for  his  services,  how  much  does  he 
collect  ?     What  per  cent  of  the  whole  tax  remains 
after  he  has  taken  his  commission  ? 

13.  If  a  tax  collector  whose  commission  is  2^> 
receives  $1200  for  his  services,  how  much  does  he 
collect  ?     What  per  cent  of  the  whole  tax  remains 
after  he  has  taken  his  commission  ? 

14.  If  a  tax  collector  whose  commission  is  l^^fc 
receives  $1000  for  his  services,  how  much  does  he 
collect  ?     What  per  cent  of  the  whole  tax  remains 
after  he  has  taken  his  commission  ? 

15.  What   sum   must  be  raised  in  order  that 
$19,600  may  remain  after  paying  a  commission  of 
2^  for  collection  ? 

16.  What   sum  must   be  raised  in  order   that 
$44,325  may  remain  after  paying  a  commission  of 

for  collection  ? 


G.]  Duties.  189 

G.  Duties. 

The  United  States  Government  needs  money 
for  — 

The  salaries  of  the  president,  senators,  congress- 
men, judges,  and  other  officers. 

Pensions  to  men  disabled  in  fighting  for  the 
preservation  of  the  Union. 

The  support  of  an  army  and  navy,  and  the 
building  and  maintenance  of  arsenals  and  forts. 

The  improvement  of  rivers  and  harbors,  so  that 
transportation  by  water  may  be  easy  and  safe. 

Interest  on  the  public  debt. 

And  for  such  other  objects  as  may  tend  to  pro- 
mote the  welfare  of  the  whole  people. 

The  largest  part  of  the  money  needed  by  the 
United  States  Government  is  obtained  by  imposing 
a  tax  on  property  that  is  brought  into  the  United 
States  from  other  countries.  Such  a  tax  is  called 
a  DUTY. 

1 .  The  duty  on  books  is  25J&  of  the  cost.    How 
much  duty  will  a  man  have  to  pay  on  a  collection 
of  books  which  will  cost  him  in  London  $1270  ? 

2.  The  duty  on  kid  gloves  is  50^o  of  the  cost. 
How  much  duty  will  an  importer  have  to  pay  on 
10  dozen  pairs  of  kid  gloves,  the  cost  of  which  in 
Paris  is  40  cents  a  pair  ? 

3.  In  1887  watches  to  the  value  of  $1,198,109 
were  imported  to  the  United  States ;  the  duty  was 

of  the  cost.     What  was  the  total  duty  ? 

4.  In  1887   30,027,670   yards  of  cotton   cloth 


190  Percentage.  [§  7. 

were  imported  into  the  United  States  at  an  aver- 
age cost  of  12  cents  a  yard  ;  the  average  duty  was 
of  the  cost.  What  was  the  total  duty  ? 


NOTE.  A  duty  based  on  the  cost,  as  in  the  last 
four  examples,  is  called  an  AD  VALOREM  *  DUTY. 

5.  The  duty  on  silk  umbrellas  is  50^?  ad  va- 
lorem.   How  much  duty  must  a  man  pay  on  25  silk 
umbrellas,  the   cost  of  which   in   Paris   is  $1.75 
apiece  ? 

6.  The  duty  on  fine  salt  is  8  cents  on  every 
hundred  pounds.    How  much  duty  must  a  man  pay 
who  imports  24,840  pounds  ? 

7.  The  duty  on  the  best  molasses  is  8  cents  a 
gallon.     How  much  duty  must  a  man  pay  who  im- 
ports 80  hogsheads  containing  63  gallons  each  ? 

8.  In  1887,  972,570  pounds  of  oat  meal  were 
imported  into  the  United  States  ;  the  duty  was  ^ 
cent  per  pound.     What  was  the  total  duty  ? 

9.  In  1887,  11,207,548  pounds  of  filberts  and 
walnuts  were  imported  into  the  United  States  ;  the 
duty  was  3  cents  a  pound  and  the  cost  was  5|-  cents 
a  pound.     What  was  the  total  duty  and  also  the 
total  cost  ? 

1O.  During  the  year  1887  there  were  imported 
into  the  United  States  33,731,463  pounds  of  rice 
at  a  cost  of  2  cents  a  pound  ;  the  duty  was  2  J  cents 
a  pound.  What  was  the  total  duty  and  also  the 
total  cost  ? 

*  Ad  valorem  means  according  to  value. 


G.]  Duties.  191 

NOTE.     A  duty  based  on  quantity,  as  in  each  of 
the  last  five  examples,  is  called  a  SPECIFIC  DUTY. 

11.  On  spirit  varnishes  there  is  a  specific  duty  of 
$1.32  a  gallon  and  an  ad  valorem  duty  of  40jfc.  How 
much  duty  must  a  man  pay  on  85  gallons  of  varnish, 
the  cost  of  which  in  London  is  $ 2.38  a  gallon  ? 

12.  During  the  year  1887  there  were  imported 
into  the  United  States  15,671  gallons  of  cologne 
water  at  a  cost  of  $15.09  a  gallon  ;  there  was  a 
specific  duty  of  $2  a  gallon  and  an  ad  valorem  duty 
of  50^.     What  was  the  total  duty  ? 

13.  During  the  year  1887  there  were  imported 
into  the  United  States  50,315  gross  of  lead  pencils 
at  a  cost  of  $1.91  a  gross  ;  there  was  a  specific 
duty  of  50  cents  a  gross  and  an  ad  valorem  duty 
of  30^.     What  was  the  total  duty  ? 

14.  During  the  year  1887  a  family  in  Boston 
bought  600  pounds  of  imported  sugar  at  an  average 
price  of  7  cents  a  pound  ;  this  sugar  cost  in  the 
countries  that  it  came  from  2.6  cents  a  pound,  and 
the  duty  was  82jfc.     How  much   less  would  the 
year's  supply  have   cost   the  family  if  there  had 
been  no  duty? 

15.  During  the  year  1887,  500,085  pounds  of 
hemp  yarn  were  imported  into  the  United  States, 
at  an  average  cost  of  12  J  cents  a  pound  ;  the  duty 
was  35^).     What  was  the  total  duty  ?     35^  is  how 
many  cents  a  pound  ?     [This  is  the  same  as  asking 
what  specific  duty  per  pound  is  equivalent  to  an 
ad  valorem  duty  of 


192  Percentage.  [§  7. 

16.  What  specific  duty  on  buckwheat  is  equiva- 
lent to  an  ad  valorem  duty  of  10^>,  when  the  cost 
is  38  cents  a  bushel  ? 

17.  In  1887  there  was  a  specific  duty  on  barley 
of  10  cents  a  bushel  ;  the  cost  price  was  60  cents 
a  bushel.     What   ad  valorem   duty  would   have 
been  equivalent  to  the  specific  duty  ?     [10  cents 
on  60  cents  (the  price  of  a  bushel)  is  £$  of  the 


18.  What  ad  valorem  duty  on  bituminous  coal 
is  equivalent  to  a  specific  duty  of  75  cents  a  ton 
when  the  cost  is  $3.08  a  ton  ? 

19.  What  ad  valorem  duty  on  raisins  is  equiva- 
lent to  a  specific  duty  of  2  cents  a  pound  when  the 
cost  is  5£  cents  a  pound  ? 

20.  What   ad   valorem   duty   on   lead    pencils 
which  cost  $1.91  a  gross  is  equivalent  to  a  specific 
duty  of  50  cents  a  gross  and  an  ad  valorem  duty 
of  30^  ?     Ans.  56^  nearly. 

21.  What   ad  valorem  duty  on   cologne  water 
which  costs  $15.09  a  gallon  is  equivalent  to  a  spe- 
cific duty  of  $2  a  gallon  and  an  ad  valorem  duty 
of 


The  United  States  Congress  is  now  (July,  1888) 
considering  the  advisability  of  reducing  the  duties 
on  imported  goods.  The  following  table  shows 
the  values  of  different  kinds  of  dutiable  property 
brought  into  the  United  States  during  the  year 
1887  ;  the  present  average  ad  valorem  duties  ;  and 
the  duties  contained  in  a  proposition  now  before 
Congress,  called  the  Mills  bill. 


G.] 


Duties. 


193 


Schedule. 

Values  of  im- 
portations in 

1887. 

Ad  valorem  duties. 

Present   %. 

Proposed  %. 

Chemicals 

$  5,050,325 

40 

22 

Earthenware   and 

glassware 

10,492,067 

66 

49 

Metals 

16,152,789 

52 

43 

Wood    and    wooden- 

ware 

889,558 

35 

29 

Sugar 

68,897,102 

82 

66 

Tobacco 

26,441 

82 

38 

Provisions 

3,235,987 

53 

43 

Cotton    and  cotton 

goods 

2,423,585 

51 

40 

Hemp,  jute,  and  flax 

goods 

17,434,514 

36 

24 

Wool  and  woollens 

42,448,127 

69 

40 

Books,  papers,  etc. 

57,298 

24 

18 

Sundries 

11,221,253 

44 

35 

Wood,  salt,  etc. 

61,672,120 

27 

00 

Wool 

18,206,987 

30 

00 

22.  a.  Find  from  the  table  given  above  the 
amount  of  the  duties  collected  in  1887  on  each 
kind  of  property  ;  find  also  the  total  amount. 

5.  Find  the  amount  that  would  have  been  col- 
lected on  each  kind  of  property  with  the  duties  as 
proposed ;  find  also  the  total  amount. 

c.  How  much  less  would  the   duties  on  sugar 
have  amounted  to  in  1887  if  the  rate  of  duty  had 
been  as  now  proposed  ? 

d.  How  much   less  would  all  the  duties  have 
amounted  to  in  1887  if  the  rates  had  been  as  now 
proposed  ? 


194  Percentage.  [§  7. 

H.  Miscellaneous. 

1.  If  I  can  buy  the  following  books  at  a  dis- 
count or  reduction  of  10$fc  (Ta^  or  ^)  from  the 
list  prices,  what  must  I  pay  for  each  ? 


LIST  PRICES. 


I.  The   Lamplighter.      By  Maria   S. 

Cummins  ....     $0.50 

II.  Frederick   the   Great.      By   T.  B. 

Macaulay 60 

III.  Handbook   of    American   Authors. 

By  O.  F.  Adams         ...         .75 

IV.  Tom  Brown  at  Rugby.    By  Thomas 

Hughes 1.00 

V.  The   American    Statesmen    Series, 

per  volume          .         .         .         .1.25 
VI.  The  Story  of  a  Bad  Boy.     By  T.  B. 

Aldrich 1.50 

VII.  Longfellow's    Poems.       Household 

Edition 1.75 

VIII.  The  Autocrat  of  the  Breakfast-Ta- 
ble.    By  O.  W.  Holmes      .         .       2.00 
IX.  Holmes'  Poems.      Household  Edi- 
tion, in  full  gilt  .         .         .         .2.25 
X.  The  Children's  Book.      By  H.  E. 

Scudder 2.50 

XI.  Whittier's  Poems.     Household  Edi- 
tion in  half  calf  .         .         .         .       3.00 
XII.  Yesterdays  with  Authors.      By  J. 

T.  Fields.     In  half  calf       .         .       3.25 
XIII.  Agassiz.      By  Elizabeth  C.  Agassiz       4.00 


H.]  Miscellaneous.  195 

XIV.  History  of  Our  Country.     By  Abby 

S.  Richardson     ....       4.50 

2.  What  will  each  book  in  the  preceding  list 
cost  at  a  discount  of  15^6  ? 

NOTE.  To  get  the  cost  of  VII.,  for  instance, 
we  may  say  I5fi  of  $1.75  =  $1.75  x  .15  =  $0.2625. 
Therefore,  the  cost  is  $1.75 -$0.2625  =  $1.4875  or 
$1.48|.  The  work  may  be  done  mentally  if  we 
note  that  15^b  or  -^-fa  =  -£fa  +  T^  —  iV  ~*~  2"  °^  lV 

TV  of  $1.75  =  $0.175  or  $0.17  J. 

J  of  ^  of  $1.75  =  .0875  or  .08|. 

Therefore  15^  =  $0.26 J;  and  the  cost  is  $1.75- 
$0.26£  or  $1.48|. 

3.  What  will  each  of  the  books  cost  at  a  dis- 
count of  20^  ?     [20^>  =  T2o°o  =  ^  or  £ .] 

4.  What  will  each  of  the  books  cost  at  a  dis- 
count of  25$fc  ?     [25^  =  |.] 

5.  What  will  each  of  the  books  cost  at  a  dis- 
count of  30f/)  ?     [30^  =  ^.] 

6.  What  will  each  of  the  books  cost  at  a  dis- 
count of  33£$  or  1  ? 

Suggestion:  After  taking  off  J  we  shall  have  ^ 
left.  Therefore  f  of  each  price  will  give  the  cost. 

7.  What  will  each  book  cost  at  a  discount  of 
40/o  ?     [The  cost  will  be  60^  or  -^  or  |.] 

8.  a.  What  will  be  the  cost  of  10  copies  of  I., 
10  of  II.,  15  of  IV.,  25  of  VI.,  and  15  of  VII.  at 
a  discount  of  £  ?  Ans.  $59.83. 

b.  If,  by  paying  cash,  I  can  get  a  discount  of  5^o 
from  the  charge  for  the  above,  what  must  I  pay  ? 

Ans.  $56.84. 


196  Percentage.  [§  7. 

NOTE.  In  this  case  I  am  said  to  get  a  discount 
of  ^  and  5J&,  which  means  that  I  may  take  off  5jfc 
of  the  result  found  by  taking  off  J. 

9.  a.  What  will  each  book  cost  at  a  discount 
of  \  and  lOJfc  ?     6.  What  at  a  discount  of  40^>  ? 

10.  Show  that  a  discount  of  J  and  10$fc  is  the 
same  as  a  discount  of  40^. 

1  1.  a.  If,  on  an  order  for  1000  copies  of  VII. 
(see  Example  1),  800  of  XIII.,  and  400  of  V., 
I  am  allowed  a  discount  of  50^,  what  must  I  pay  ? 
6.  If  I  am  allowed  a  discount  of  40$fc  and  10^, 
what  must  I  pay  ? 

12.  What  Jb  is  a  discount  of  40^fc  and  10#  ? 

13.  a.  How  much  does  a  bookseller  make  on  a 
$3.00  book  that  he  buys  at  a  discount  of  ^  and 
sells  to  you  at  a  discount   of   20*fc  ?     What  per 
cent   does    he    make   on    his    investment  ?      [He 
makes  40  cents  on  $2.00,  or  20  cents  on  $1.00, 
or  20^.] 

6.  If  he  buys  the  book  at  a  discount  of  40^ 
and  sells  it  to  you  at  a  discount  of  ^,  what  is  his 
profit  ?  what  per  cent  ? 

Solution:  On  the  $1.80  (180  cents)  that  he 
pays  for  the  book  he  makes  a  profit  of  20  cents  ; 
a  profit  of  1  cent  would  be  T^  of  the  cost  ;  his 
profit,  then,  is  -^fa  or  ^  of  the  cost  ;  and  this, 
reckoned  by  per  cent  (that  is,  by  the  hundred),  is 
1  of  100  or  ll^Jfc.  We  get  the  same  result  if  we 
reduce  -J-  to  hundredths,  as  in  decimals.  Thus  £  = 


c.  Find  HJJfe  of  $1.80. 


H.]  Miscellaneous.  197 

14.  A  bookseller  sold  a  shop- worn  book  for  85 
cents  that  cost  him  $1.00  ;  what  per  cent,  did  he 
lose  ?  Ans.  15^. 

15.  A  man  sold  a  horse  for  $  130  that  cost  him 
$150  ;  what  per  cent  did  he  lose  ?          Ans.  13£j6. 

16.  A  merchant  sold  goods  at  auction  for  $3250 
that  cost  him  $5000  ;  what  per  cent  did  he  lose  ? 

17.  A  merchant  sold   a  quantity  of   goods  for 
$273.00,  by  which  he  gained  10  per  cent  on  the 
cost.     What  was  the  cost  ? 

Suggestion:    10  per  cent,   is  y1^  of  the  cost. 
Consequently  $273.00  must  be  {±$  of  the  cost. 

18.  A  merchant  sold  a  quantity  of  goods  for 
$135.00,  by  which  he  gained  13  per  cent.     How 
much  did  the  goods  cost,  and  how  much  did  he  gain? 

19.  A  merchant  sold  a  quantity  of   goods  for 
$3875,  by  which   he   gained  65  per  cent.     How 
many  dollars  did  he  gain  ? 

20.  A  merchant  sold  a  quantity  of  goods  for 
$983.00,  by  which  he  lost  12  per  cent.     How  much 
did  the  goods  cost,  and  how  much  did  he  lose  ? 

NOTE.      If  he  lost  12  per  cent.,  that  is   T^, 
he  must  have  sold  it  for  -ffa  of  what  it  cost  him. 

21.  A    farmer    sold    3   cows   for   $248.37,  by 
which   he*  lost  25  per  cent.     How  much  did  the 
cows  cost  him,  and  how  much  did  he  lose  ? 

22.  A  merchant    sold  a  quantity  of  goods  for 
$87.00  more  than  he  gave  for  them,  by  which  he 
gained  13  per  cent  of  the  cost.      What  did  the 
goods  cost  him,  and  how  much  did  he  sell  them 
for? 


198  Percentage.  [§  7. 

NOTE.  Since  13  per  cent,  is  T^,  |87  must  be 
^^  of  the  first  cost. 

23.  A  merchant  sold  a  quantity  of  goods  for 
$4$. 00  more  than  they  cost,  and  by  so  doing  gained 
20  per  cent.     How  much  did  the  goods  cost  him  ? 

24.  A  merchant  sold  a  quantity  of  goods  for 
$137.00  less  than  they  cost  him,  and  by  doing  so 
lost  23  per  cent.     How  much  did  the  goods  cost, 
and  how  much  did  he  sell  them  for  ? 

25.  A  man  having  put  a  sum  of  money  at  inter- 
est at  6  per  cent,  at  the  end  of  one  year  received 
13  dollars  for  interest.     What  was  the  principal  ? 

NOTE.  Since  6  per  cent  is  -j  ^  of  the  whole,  13 
dollars  must  be  ^077  °f  the  principal. 

26.  What  sum  of  money  put  at  interest  for  1 
year  will  gain  $57,  at  6  per  cent  ? 

27.  A  man  put  a  sum  of  money  at  interest  for 
1  year,  at  6  per  cent,  and  at  the  end  of  the  year 
he  received  for  principal  and  interest  237  dollars. 
What  was  the  principal  ? 

NOTE.  Since  6  per  cent  is  y§-^,  if  this  be 
added  to  the  principal  it  will  make  |§f ,  therefore 
$237  must  be  \^  of  the  principal. 

28.  What  sum  of  money  put  at  interest  at  6  per 
cent  will  gain  $53  in  two  years  ? 

29.  What  sum  of  money  put  at  interest  at  6 
per  cent  will  gain  $97  in  one  year  and  6  months  ? 

30.  What  sum  of  money  put  at  interest  at  6  per 
cent  will  amount  to  $394  in  1  year  and  8  months  ? 

31.  What  sum  of  money  put  at  interest  at  7 
per  cent  will  amount  to  $183  in  one  year  ? 


H.]  Miscellaneous.  199 

32.  What  sum  of  money  put  at  interest  at  8  per 
cent  will  amount  to  $137  in  2  years  and  6  months  ? 

33.  Suppose  I  owe  a  man  $287  to  be  paid  in 
one  year  without  interest,  and  I  wish  to  pay  it 
now;    how  much  ought    I   to  pay  him,  when  the 
usual  rate  is  6  per  cent.  ? 

34.  A  man  owes  $847  to  be  paid  in  6  months, 
without  interest ;  what  ought  he  to  pay  if  he  pays 
the  debt  now,  allowing  money  to  be  worth  6  per 
cent  a  year  ? 

35.  A  merchant  being  in  want  of  money  sells  a 
note  of  $100,  payable  in  8  months,  without  inter- 
est.    How  much  ready  money  ought  he  to  receive, 
when  the  yearly  interest  of  money  is  6  per  cent.  ? 

36.  According  to  the  above  principle,  what  is 
the  difference  between  the  interest  of  $100  for  1 
year  at  6  per  cent  and  the  discount  of  it  for  the 
same  time  ? 

37.  What  is  the  difference  between  the  interest 
of  $500  for  4  years  at  6  per  cent  and  the  discount 
of  the  same  sum  for  the  same  time  ? 

38.  At  an  arithmetic   examination    a  boy  did 
correctly  only  |  of  the  work  required  ;  what  per 
cent  of  the  work  did  he  do  ? 

39.  At  a  civil  service  examination  a  man  got  58 
marks  out  of  80  ;  what  was  his  per  cent  ? 

40.  On  Jan.  1,  1887,  a  horse-car  conductor  who 
had  been  receiving  $50   a  month  had  his  wages 
lowered  10J6.     On  Feb.  1  his  request  for  10^  more 
pay  than  he  was  then  receiving  was  granted.     How 
much   a  month   did   he  receive   after   these   two 
changes  ? 


200  Percentage.  [§  7. 

41.  A  man  who  had  11000  lost  20^  of  it  in  a 
trade  ;  he  then  invested  what  he  had  left  in  flour  ; 
what  per  cent  did  he  make  on  the  flour  if  he  sold 
it  for  $1000  ? 

42.  Just  before  Christmas  a  jeweler  increased 
by  lOjfc  the  selling  prices  of  the  following  articles : 

Selling  prices. 

Waltham  silver  watches  .  .  .  $25.00 
Gold  watch  chains  ....  10.00 
Small  clocks  .  .  .  .  .  2.50 

What  did  the  selling  prices  then  become  ? 

43.  After  Christmas  he  [see  Example  42]  low- 
ered the  new  selling  prices  by  lOyfc  ;  what  did  the 
selling  prices  then  become  ? 

44.  A  superintendent  of  schools  called  for  writ- 
ten answers  to  Example  40  from  the  pupils  in  each 
of  6  schools  under  his   charge.      The   following 
table  shows  the  number  of  pupils  in  each  school, 
and  the  number  who  answered  the  question  cor- 
rectly. 

Complete  the  table  by  entering  in  the  fourth 
column  the  per  cent  of  the  pupils  in  each  school 


who  gave  correct  answers. 


School. 

No.  1 

Whole  No. 
58 

No.  answering 
correctly. 

14 

No.  2 

64 

12 

No.  3 

38 

16 

No.  4 

42 

12 

No.  5 

75 

13 

No.  6 

80 

20 

answering 
correctly. 


H.]  Miscellaneous.  201 

45.  A  man's  net  *  income  from  his  property  is 
$5000  ;  how  much  property  has  he  if  it  earns  5^> 
a  year,  subject  to  no  expenses  or  charges  except  a 
tax  of  l/o  ? 

46.  A  man  bought  a  house  for  $1200  ;  he  spent 
$400  for  repairs,  paid  a  tax  of  \ffl  of  the  cost,  and 
at  the  end  of  a  year  sold  it  for  $1650.     Would  he 
have  made  more  or  less,  and  how  much,  if  he  had 
loaned  his  money  at  5j#,  and  had  paid  a  tax  of 

? 

47.  A  merchant  buys  furniture  at  a  discount  of 
and  10^  from  the  list  prices,  and  sells  it  at  a 

discount  of  10^  and  5fi.    What  per  cent  of  the 
cost  is  his  profit  ? 

48.  What  is  the  difference  on  a  bill  of  $425 
between  a  discount  of  50^fc  and  a  discount  of  40yfc 
and  10#  ? 

49.  80  marks  were  to  be  given  for  perfect  an- 
swers to  all  the  questions  on  an  examination  paper 
in  arithmetic.     The  following  are  the  marks  of  the 
different  pupils  in  a  class  of  6  :  80,  65,  58,  42,  38, 
and  25.     What  per  cent  did  each  get  ? 

50.  By  selling  a  house  for  $2340  a  man  lost  10$6 
of  the  cost ;  what  should  he  have  sold  it  for  to 
make  10#  ? 

51.  A  collector  deducts  his  commission  of  2^? 
from  a  bill ;  the  balance  is  $1960.    What  was  the 
biU? 

52.  A  boy  buys  chestnuts  at  $2.50  a  bushel,  and 
sells  them  at  5  cents  a  pint.     What  per  cent  does 
he  make  ? 

*  What  remains  after  deducting  all  charges  and  expenses. 


202  Percentage.  [§  7. 


53.  A  stationer  marks  his  writing  paper  at 
above  cost.     What  discount  must  he  allow  a  cus- 
tomer in  order  to  make  a  profit  of  lOjfc  ? 

Solution  :  Paper  that  costs  $1.00  is  marked 
$1.25;  15  cents  or  T\^  from  this  leaves  $1.10, 
which  gives  a  profit  of  lOjfc  on  the  cost.  There- 
fore from  the  marked  price  he  must  make  a  dis- 

count of  VA  =  i$  =  lffi>  =  l2fi  A™> 

54.  The  marked  prices  of  a  jeweler's   articles 
are  40^)  above  the  cost  prices  ;  how  much  discount 
can  he  make  from  the  marked  prices  and  make  a 
profit  of  20*fc  on  the  cost  ? 

55.  What  Jo  of  the  cost  will  his  profit  be  if  he 
makes  a  discount  of  20<fc  from  the  marked  prices  ? 

56.  A  wholesale  bookseller  buys  a  lot  of  books 
from  a  publisher  at  a  discount  of  40^  and  sells 
them  again  at  a  discount  of  ^  and  5yfc.    What  per 
cent  of  the  cost  is  his  profit? 

57.  Which   of  the  two   following   investments 
will  yield  the  greater  net  income  in  a  town  where 
the  tax  is  1  \°/o  ?     How  much  greater  ? 

A.  $10,000  invested  in  U.  S.  4jfc  bonds  at  120. 
[Such  bonds  are  exempt  from  taxation.] 

B.  $10,000  invested  in  railroad  stock  at  110, 
which  pays  a  dividend  of  5^o  and  is  subject  to 
taxation. 

NOTE.  The  rest  of  the  questions  in  this  section 
are  taken  from  examination  papers  set  for  admis- 
sion to  Harvard  College. 

58.  At  what  rate  per  cent  must  $370  be  put  on 
interest  to  gain  $55.50  in  three  years  ? 


H.]  Miscellaneous.  203 

59.  Find  the  interest  on  176.72  from  April  18, 
1852,  to  January  26,  1855,  at  6  per  cent. 

60.  What  principal,  at  6  per  cent,  will  amount 
to  $360,585  in  16  months  ? 

61.  If  $50  gain  $5.60  in  3  yrs.  6  mos.  at  simple 
interest,  what  is  the  rate  per  cent  ? 

62.  What  is  the  amount  of  $5216.75  from  Jan- 
uary 21,  1860,  to  July  3,  1863,  at  8  per  cent,  com- 
pound  interest  ? 

63.  Find  the  amount  of  $1000  for  2  yrs.  2  mos. 
12  ds.,  at  6  per  cent,  compound  interest. 

64.  Calculate  the  date  at  which  a  sum  of  $450, 
which  was  put  at  simple  interest  at  8  per  cent, 
December  30,  1797,  amounted  to  $642.30. 

65.  Calculate  the   date   at  which   the  sum  of 
$234,  put  at  simple  interest  at  9  per  cent,  Octo- 
ber 25,  1798,  amounted  to  $351. 

66.  Find  the  simple  interest  on  $1000  for  5  yrs. 
4  mos.  and  15  dys.  at  20  per  cent.     To  how  much 
will  $1000  amount  in  4  yrs.  at  compound  interest, 
at  20  per  cent  ? 

67.  A  certain  sum  of  money  was  put  at  simple 
interest  at  9  per  cent,  December  21,  1790.     At 
what  date  did  it  become  tripled  ? 

68.  I   owe   three   notes    bearing  interest  from 
date  :  the  first,  dated  June  1, 1866,  is  for  $450.00  ; 
the  second,  dated  Dec.  17,  1866,  is  for  $750.00  ; 
the  third,  dated  March  15,  1867,  is  for  $600.00. 
I  wish  to  substitute  for  these  a  single  note  for 
$1800.00  :  what  should  be  the  date  of  it  ? 


204  Percentage.  [§  7. 

69.  A  certain  bank  declares  a  semi-annual  divi- 
dend of  4  per  cent :  what  can  I  afford  to  pay  for 
its  shares  if  I  wish  to  get  6  per  cent  a  year  for  my 
money  ? 

70.  What  is  the  interest  of  $875.26  from  Octo- 
ber 10, 1866,  to  July  10,  1868,  at  7^  per  cent? 

71.  The  sum  £46  6s.  Sd.  was  put  at  interest  at 
4  per  cent  on  the  20th  June,  1868.     Required  the 
amount  on  the  5th  of  May,  1875. 

72.  At  what  rate  of  simple  interest  will  $2500 
amount  in  3  years  to  $3250? 

73.  Bought  $1500  worth  of  goods,  half  on  6 
months'  and  half  on  9  months'  credit.    What  sum, 
at  7  per  cent  interest,  paid  down,  would  discharge 
the  whole  bill  ? 

74.  The   capital    stock  of    a  certain   bank   is 
$500,000,  and  directors  have  declared  a  dividend 
of  4  per  cent.     The  sum  set  aside  from  the  profits 
to  meet  this  dividend  is  subject  to  a  revenue  tax 
of  5  per  cent.     What  sum  must  be  set  aside  in 
order  that  the  stockholder  may  receive  a  dividend 
of  4  per  cent  on  his  stock  ? 

75.  Find  the  interest  on  one  pound  sterling  at  5 
per  cent  for  one  year  ;  for  one  month. 

76.  My  agent  sells  for  me  2000  yards  of  cloth 
at  24  cents  a  yard.     He  allows  the  purchaser  five 
per  cent  discount  for  cash,  and  charges  me  2|  per 
cent  on  the  cash  receipts.     How  much  money  does 
he  pay  over  to  me  ? 

77.  What  is  the  amount  of  $340  at  8  per  cent 


H.]  Miscellaneous.  205 

for  1  year,  3  months,  the  interest  being  compounded 
semi-annually  ? 

78.  I  buy  one  fifth  of  an  acre  of  land  for  $2178. 
For  how  much  a  square  foot  must  I  sell  it,  in  order 
to  gain  twenty  per  cent  of  the  cost  ? 

79.  Having  purchased  an  acre  of  land,  I  sell 
from  it  a  rectangular  lot,  121  yds.  long  and  25  yds. 
wide,  for  what  the  whole  acre  cost  me.     What  per 
cent  do  I  gain  on  the  land  thus  sold  ? 

80.  A  collector  who  charges  8  per  cent  com- 
mission  on  what   he   collects    pays   me   $534.75. 
What  amount  does  he  collect  ? 


SECTION  VIII. 


SQUARE  AND  SOLID  MEASURES. 

A.     Square  Measure. 

IN  measuring  surfaces,  such  as  land,  etc.,  a 
square  is  used  as  the  measure  or  unit.  A  square 
is  a  figure  with  four  equal  sides, 
and  with  four  equal  corners  or 
angles.  A  square  is  used  be- 
cause it  is  a  more  convenient 
measure  than  a  figure  of  any 
other  form.  The  figure  ABCD 
is  a  square ;  the  sides  are  each 
one  inch,  consequently  it  is  called 
a  square  inch.  A  similarly  shaped  figure  one  foot 
long  and  one  foot  wide  is  called  a  square  foot ;  a 
similarly  shaped  figure  one  yard  long  and  one  yard 
wide  is  called  a  square  yard,  etc. 

1  inch  wide  con- 
tains 1  square 
inch,  how  many 
square  inches 
does  a  figure  1 
inch  wide  and  2 
inches  long  con- 
tain?  How 


1.  If  a  figure  1  inch  long  and 


A.]  Square  Measure.  207 

many  square  inches  does  a  figure  1  inch  wide  and 
3  inches  long  contain  ?  1  inch  wide  and  4  inches 
long  ?  1  inch  wide  arid  5  inches  long  ?  1  inch  wide 
and  7  inches  long  ? 

2.  How  many  square  inches  are  there  in  a  fig- 
ure 8  inches  long  and  1  inch  wide  ?  8  inches  long 
and  2  inches  wide?  8  inches  long  and  3  inches 
wide  ?  8  inches  long  and  4  inches  wide  ?  8  inches 
long  and  5  inches  wide  ?  8  inches  long  and  8  inches 
wide? 

3.  If  a  figure  1  foot  wide  and  1  foot  long  con- 
tains 1   square  foot,  how  many  square  feet  does 
a  figure  1  foot  wide  and  2  feet  long  contain  ?  1 
foot  wide  and  3  feet  long  ?  1  foot  wide  and  5  feet 
long?  1  foot  wide  and  9  feet  long?  1  foot  wide 
and  15  feet  long  ? 

4.  How  many  square  feet  are  there  in  a  figure 
9  feet  long  and  1  foot  wide?    9  feet  long  and  2 
feet  wide?    9  feet  long  and  3  feet  wide?  9  feet 
long  and  5  feet  wide  ?  9  feet  long  and  7  feet  wide  ? 
9  feet  long  and  9  feet  wide  ? 

5.  How  many  square  inches  does  a  figure  13 
inches  long  and   1  inch  wide  contain  ?  13  inches 
long  and  2  inches  wide?  13  inches  long  and  3 
inches  wide  ?  13  inches  long  and  8  inches  wide  ? 

6.  How  many  square  feet  does  a  figure  16  feet 
long  and  1  foot  wide  contain  ?  16  feet  long  and  2 
feet  wide  ?  16  feet  long  and  3  feet  wide  ?  16  feet 
long  and .  5  feet  wide  ?    16  feet  long  and  8  feet 
wide?  16  feet  long  and  13  feet  wide  ? 

NOTE.    In  the  preceding  examples  supply  yards, 


208  Square  and  Solid  Measures.          [§  8. 

rods,  and  miles,  instead  of  inches  and  feet,  and 
find  the  result. 

NOTE.  A  figure  with  four  sides,  which  has  all 
its  angles  alike  or  right  angles,  is  called  a  rectan- 
gle ;  a  rectangle  is  called  a  square  when  all  the 
sides  are  equal. 

7.  What  rule  can  you  make  for  finding  the 
number  of  square  inches,  feet,  yards,  etc.,  in  any 
rectangular  figure  ? 

8.  How  many  square  feet  are  there  in  the  floor 
of  a  room  18  feet  long  and  13  feet  wide  ? 

9.  How  many  square  feet  are  there  in  a  piece 
of  land  143  feet  long  and  97  feet  wide  ? 

10.  How  many  square  rods  are  there  in  a  piece 
of  land  28  rods  long  and  7  rods  wide  ? 

11.  A  piece  of  land  that  is  20  rods  long  and  8 
rods  wide   (or  in  any  other  form  containing  the 
same  surface),  is  called  an  acre.    How  many  square 
rods  are  there  in  an  acre  ? 

12.  How  wide  must  a  piece  of  land  be  that  is 
16  rods  long,  to  contain  an  acre  ? 

13.  How  many   square  inches    are   there  in  a 
square  foot ;  that  is,  in  a  figure  that  is  12  inches 
long  and  12  inches  wide  ? 

14.  How  long  must  a  rectangle  be  that  is  8 
inches  wide,  in  order  to  contain  a  square  foot  ? 

15.  How  many  square  feet  are  there  in  a  square 
yard? 

16.  How  many   square   yards   are   there   in   a 
square  rod  ? 

17.  How  many  square   inches   are   there  in  a 
square  yard  ? 


A.]  Square  Measure.  209 

18.  Find  the  numbers  needed  to  complete  the 
following  table  :  — 

Square  Measure. 

144  square  inches          =1  square  foot  (sq.  ft.). 
square  feet  =1  square  yard  (sq.  yd.). 

square  yards  or  j  =1  e  ^  (      rd  } 

square  feet          ) 

= 


square  rods  =1  acre. 


19.  How   many  square   inches   are  there  in  a 
square  rod  ? 

20.  How  many  square  yards  are  there  in  an 
acre  ? 

21.  How  many  square  inches  are  there  in  an 
acre  ?  ^ 

22.  How  many  square  feet  are  there  in  1728 
square  inches  ? 

23.  How*  many  acres  are  there  in  286  square 
rods  ? 

24.  How  many  acres  are  there  in  201,283,876 
square  inches  ? 

25.  How  many  square  rods  are  there  in  a  square 
mile  ? 

26.  How   many   acres   are   there   in   a  square 
mile? 

27.  The  whole  surface  of  the  globe  is  estimated 
at  about  198,000,000  square  miles.     How  many 
acres  are  there  on  the  surface  of  the  globe  ? 

28.  How  many  square  inches  are  there  on  one 
side  of  a  board  15  inches  wide  and  11  feet  long  ? 
How  many  square  feet  ? 


210  Square  and  Solid  Measures.  [§  8. 

29.  How  many  acres  are  there  in  a  piece  of  land 
183  rods  long  and  97  rods  wide  ? 

30.  How  many  tomato  plants  can  be  set  in  a 
garden  60  feet  by  21  feet,  if  a  square  yard  is 
allowed  to  each  plant  ? 

31.  How  many  six-acre  fields  can  be  made  out  of 
a  piece  of  land  one  mile  long  and  half  a  mile  wide, 
and  how  many  acres  will  be  left  over  ? 

32.  How  many  acres  are  there  (a)  in  a  lot  of 
land  80  rods  square,  and  (6)  in  a  lot  of  land  160 
rods  by  40  rods  ? 

At  |3  a  rod,  how  much  more  will  it  cost  to  fence 
the  second  lot  than  the  first  lot  ?  Illustrate  by  a 
diagram  of  each  lot.  [If  a  line  of  any  convenient 
length  be  taken  to  represent  160  rods,  then  a  line 
half  as  long  will  represent  80  rods,  and  a  line  one 
fourth  as  long  will  represent  40  rods.] 

33.  From  a  lot  of  land  60  rods  square  60  square 
rods  were  sold ;  what  was  the  value  of  the  remain- 
der at  1160  an  acre? 

34.  Show  that  a  garden  40  feet  square  contains 
the  same  number  of  square  feet  as  a  garden  20 
feet  by  80  feet ;   at  30  cents  a  foot,  how  much 
more  would  it  cost  to  fence  the  second  than  the 
first  ?     Illustrate  by  a  diagram. 

35.  What  is  the  difference  in  square  rods  be- 
tween 2  square  rods  of  land  and  a  lot  of  land  2 
rods  square?  between  3  square  rods  and  3  rods 
square  ?  between  4  square  rods  and  4  rods  square  ?  ' 
between  5  square  rods  and  5  rods  square  ?  between 
10  square  rods  and  10  rods  square? 


A.]  Square  Measure.  211 

36.  How  many  square  yards  of  plastering  are 
there  in  the  ceiling  of  a  room  15  ft.  by  18  ft.  ? 

37.  How  many  square  yards  of  plastering  are 
there  in  a  room  24  feet  long,  18  feet  wide,  and  10 
feet  high,  if  the  doors  and  windows  take  up  150 
square  feet? 

38.  What  would  it  cost  to  plaster  the  ceiling  of 
a  room  33  ft.  by  15  ft.  at  15  cents  a  square  yard  ? 

39.  What  would  it  cost  to  plaster  a  wall  15  feet 
long  and  9^  feet  high,  with  an  18-inch  base  board, 
at  12  cents  a  square  yard  ? 

40.  How  many  square  yards  are  there  in  the 
floor,  walls,  and  ceiling  of  a  room  20  feet  long,  15 
feet  wide,  and  10  feet  high  ? 

41.  A  man  bought  a  piece  of  land  1000  feet 
long  and  400  feet  wide  at  20  cents  a  square  foot ; 
how  much  did  the  land  cost  him  ? 

In  order  to  make  the  land  available  for  house 
lots,  he  put  two  streets,  each  50  feet  wide,  through 
the  centre,  one  in  the  direction  of  the  width  and 
the  other  in  the  direction  of  the  length.  How 
many  square  feet  did  these  streets  occupy,  and  how 
many  square  feet  remained  for  house  lots  ?  Illus- 
trate by  a  diagram. 

How  much  did  the  man  make  on  his  investment 
if,  after  spending  $5000  in  the  construction  of  the 
streets,  he  sold  the  remainder  at  30  cents  a  square 
foot? 

42.  How  many  square  feet  are  there  in  a  gravel 
walk  4  feet  wide,  which  runs  around  the  outside  of 
a  garden  40  feet  by  28  feet  ?     Illustrate  by  a  dia- 
gram. 


212  Square  and  Solid  Measures.         [§  8. 

43.  How  wide  is  a  piece  of  land  80  rods  long, 
which  contains  10  acres  ? 

44.  How  long  is  a  board  18  inches  wide  which 
contains  3  square  yards  ? 

45.  How  many  tiles  7  inches  square  will  be  re- 
quired to  cover  a  floor  19  ft.  3  in.  by  13  ft.  5  in.  ? 

46.  The  base  of  the  Great  Pyramid  of  Egypt  is 
a  square  of  764  feet  on  each  side.     Find  the  num- 
ber of  acres  of  ground  covered  by  it. 

47.  a.  How  many  square  inches  are  there  in  a 
yard  of  carpeting  that  is  2  ft.  3  in.  wide  ?     6.  How 
many  breadths  or  strips  of  such  carpeting  would  it 
take  to  cover  a  floor  18  feet  wide  ?     c.  How  many 
yards  of  such  carpeting  would  it  take  to  cover  a 
floor  24  feet  long  and  18  feet  wide  ? 

48.  a.  How  many  breadths  of  carpeting  f  of  a 
yard  wide  would  be  required  for  a  room  35  feet 
long  and  17  feet  wide,  if  the  breadths  should  run 
lengthwise  of  the  room  ?     6.   What  portion  of  one 
of  the  breadths  would  have  to  be  turned  under  or 
cut  off  ?      c.  How  many  yards  of  carpeting  would 
be  required  if  nothing  should  be  lost  in  matching 
the  pattern  ?     d.  What  would  be  the  cost  of  the 
carpet  at  $1.75  a  yard? 

e.  What  would  be  the  difference  in  cost  if  the 
breadths  should  run  across  the  room  ? 

49.  a.  How  should  the  breadths  run  in  a  room 
22  feet  long  and  19  feet  wide  in  order  to  use  the 
least  amount  of  carpeting  that  is  f  of  a  yard  wide  ? 
6.  What  would  be  the  cost  of  enough  carpeting 
for  such  a  floor  at  $2.25  a  yard  ? 


A.]  Square  Measure.  213 

50.  What  would  it  cost  to  carpet  a  room  24  ft. 
by  15  ft.  with  carpeting  27  inches  wide  at  $1.50  a 
yard,  if  the  breadths  should  run  lengthwise  of  the 
room? 

NOTE.  The  patterns  of  most  carpets  are  shown 
off  better  by  having  the  breadths  run  lengthwise 
of  the  room  rather  than  across  it. 

51.  On  June  14,  1888,  Professor  Otto  Lugger, 
of   the  State  University  of   Minnesota,   reported 
that  there  were  on  100  square  miles  in  the  region 
about  Perham,  Minn.,  12  Rocky  Mountain  locusts 
to  the  square  foot.     How  many  locusts  were  there 
on  the  entire  100  square  miles  ? 

[There  are  640  acres  in  a  square  mile.] 

52.  Reduce  .375  ft.  to  inches.     Ans.  4.5  in. 

53.  Reduce  .375  sq.  ft.  to  sq.  in.     Ans.  54  sq.  in. 

54.  Test  the  accuracy  of  the  answers  to  the  last 
two  examples  by  changing  the  inches  back  to  deci- 
mals of  a  foot. 

55.  Reduce  .02  of  an  acre  to  square  feet. 

56.  Change  23351  sq.  ft.  to  the  decimal  of  an 
acre,  carrying  out  the  result  to  three  places  of  deci- 
mals. 

57.  How  much  is  land  worth  an  acre  that  sells 
at  40  cents  a  square  foot  ? 

58.  If  a  man  buys  an  acre  of  land  for  $1000, 
for  how  much  must  he  sell  it  per  square  foot  in 
order  to  make  $1500  on  his  investment  ?    Find  the 
result  to  the  nearest  tenth  of  a  cent. 

Work  out  the  next  four  examples  by  first  changing  the  inches 
in  each  case  to  the  decimal  of  a  foot.  Carry  out  each  result  to 


214  Square  and  Solid  Measures.          [§  8. 

to  the  nearest  hundredth  only.  [If  asked  to  express  the  number 
145.6875,  for  example,  to  the  nearest  hundredth,  we  should  call 
it  145.69  rather  than  145.68.] 

59.  How  many  square  feet  are  there  on  one  side 
of  a  board  9  inches  wide  and  15  ft.  3  in.  long  ? 

60.  How  many  square  feet  are  there  in  a  floor 
14  ft.  7  in.  wide  and  19  ft.  4  in.  long  ? 

61.  How  many  square  feet  of  surface  are  there 
on  one  side  of  a  board  1  ft.  8  in.  wide  and  17  ft. 
10  in.  long? 

62.  a.  What  is  the  average  width  of  a  board 
that  is  16  inches  wide  at  one  end  and  21  inches 
wide  at  the  other  end  ?     b.  How  many  square  feet 
of  surface  are  there  on  one  side  of  such  a  board,  if 
the  length  is  6  ft.  8  in.  ? 

63.  On  almost  every  map  is  given,  in  some  con- 
venient place  near  the  margin,  a  scale  of  miles,  by 
the  aid  of  which  we  may  find  how  many  miles  are 
represented  by  any  distance  on  the  map.      The 
scale  of  miles  on  a  map  of  the  United  States  issued 
by  the  Government  in  1881  shows  that  an  inch  on 
the  map  stands  for  40  miles.     How  many  square 
miles  does  a  square  inch  on  the  map  stand  for  ? 
The  State  of  Colorado  is  represented  on  this  map 
by  a  four-sided  figure  which,  by  the  aid  of  a  tape 
measure,  I  find  to  be  7  inches  high,  9{^  inches 
wide  at  the  top,  and  9y3g  inches  wide  at  the  bottom. 
What  is  the  average  width  ?     How  many  square 
inches  are  there  in  the  figure  ?     How  many  square 
miles,  then,  should  there  be  in  Colorado  ?     Com- 
pare the  result  with  the  size  of  Colorado  as  given 


R] 


Solid  Measure. 


215 


on  page  29,  and  find  what  per  cent  your  error  is 
of  the  correct  amount.  Can  you  think  of  any  rea- 
sons why  we  should  not  expect  absolutely  accurate 
results  from  this  method  of  measurement  ? 

64.  Refer  to  the  largest  map  of  the  United 
States  that  you  have,  and,  by  the  aid  of  the  scale 
of  miles  on  it,  find  as  accurately  as  you  can  the 
distance  in  a  straight  line  from  Boston  to  San 
Francisco ;  from  St.  Paul  to  New  Orleans ;  from 
Washington  to  Chicago.  Find,  also,  the  number 
of  square  miles  in  Wyoming  Territory. 


B.    Solid  Measure. 

To  measure  solid  bodies  such  as  wood,  earth, 
stone,  etc.,  it  is  necessary  to  use  a  measure  that  has 
length,  breadth,  and  thickness  (height  or  depth). 
For  this  purpose  a  measure  is  used  in  which  the 
length,  breadth,  and  thickness  are  all  equal  to  one 
another,  and  the  corners  or 
angles  are  equal  to  one  an- 
other ;  such  a  measure  is 
called  a  CUBE.  A  solid  (like 
the  one  represented  in  the 
margin),  one  inch  long,  one 
inch  wide,  and  one  inch 
thick,  with  all  its  angles 
equal  to  one  another,  is 


called  a  cubic  inch ;  a  like  solid,  one  foot  long, 
one  foot  wide,  and  one  foot  thick,  is  called  a  cubic 
foot. 


216  Square  and  Solid  Measures.          [§  8. 

1.  How  many  edges  has  a   cube?    how  many 
corners  ?  how  many  faces  or  sides  ? 

2.  How  many  square  inches  of  paper  would  be 
needed   to  cover   the  outside  of  a  cubic  inch  of 
wood  ?  how  many  to  cover  the  outside  of  a  cubic 
foot  of  wood  ? 

3.  If  a  solid  1  inch  wide,  1  inch  thick,  and  1 
inch  long  contains  1  cubic  inch,  how  many  cubic 
inches  are  there  in  the  solid  represented   below, 
which  is  1  inch  wide,  1  inch  thick,  and  2  inches 
long? 


4.  a.  How  many  cubic  inches  are  there  in  a 
solid  1  inch  wide,  1  inch  thick,  and  3  inches  long? 
how  many  in  a  solid  of  the  same  width  and  thickness 
and  4  inches  long?  5  inches  long?  8  inches  long? 

b.  How  many  square  inches  of  surface  are  there 
on  each  of  these  solids  ? 

5.  How  many  cubic  inches  are  there  in  a  solid 
that  is  1  foot  long,  1  inch  thick,  and  1  inch  wide  ? 
How  many  cubic  inches  are  there  in  a  solid  of  the 
same  length  and  thickness  that  is  2  inches  wide? 
3  inches  wide  ?   4  inches  wide  ?   5   inches  wide  ? 
8  inches  wide? 


B.]  Solid  Measure.  217 

6.  a.  How  many  cubic  inches  are  there  in  a 
solid  that  is  2  inches  long,  2  inches  wide,  and  1 
inch  thick  ?   2  inches  long,  2  inches  wide,  and  2 
inches  thick  ? 

b.  How  many  square  inches  of  surface  are  there 
on  each  of  these  solids  ? 

7.  How  many  cubic  inches  are  there  in  a  solid 
that  is  4  inches  long,  3  inches  wide,  and  1  inch 
thick  ?  how  many  in  a  solid  of  the  same  length  and 
width  and  2  inches  thick  ?  of  the  same  length  and 
width  and  3  inches  thick? 

8.  How  many  cubic  inches  are  there  in  a  solid 
that  is  10  inches  long,  8  inches  wide,  and  1  inch 
thick ;  how  many  in  a  solid  of  the  same  length  and 
width  and  2  inches  thick  ?  3  inches  thick  ?  5  inches 
thick  ?  7  inches  thick  ? 

9.  How  many  cubic  inches  are  there  in  a  solid 
that  is  18  inches  long,  13  inches  wide,  and  1  inch 
thick  ?  how  many  in  a  solid  of  the  same  length 
and  width  and  5  inches  thick  ?  of  the  same  length 
and  width  and  11  inches  thick? 

10.  What  rule  can  you  make  for  finding  the 
number  of  cubic  inches  or  feet  in  any  solid  with 
eight  equal  corners  or  angles  ? 

11.  a.  How  many  cubic  inches  are  there  in  a 
solid  12  inches  long,  12  inches  wide,  and  12  inches 
thick,  that  is,  in  a  cubic  foot  ? 

b.  How  many  square  inches  of  surface  are  there 
on  such  a  solid  ? 

12.  How  many  cubic  feet  are  there  in  a  solid 
that  is  a  yard  long,  a  yard  wide,  and  a  yard  thick? 


218  Square  and  Solid  Measures.          [§  8. 

13.  A  pile  of  wood  that  is  8  feet  long,  4  feet 
wide,  and  4  feet  high  (or  in  any  other  form  con- 
taining the  same  quantity  of  wood),  is  called  a 
CORD  of  wood.     How  many  cubic  feet  are  there 
in  a  cord  ? 

14.  Find  the  numbers  needed  to  complete  the 
following  table :  — 

cubic  inches  =  1  cubic  foot  (cu.  ft.), 
cubic  feet     =  1  cubic  yard  (cu.  yd.), 
cubic  feet     =  1  cord. 

15.  How  many  cubic  inches  are  there  in  a  cord  ? 

16.  How  many  cubic  feet  are  there  in  509,216 
cubic  inches  ?  how  many  cubic  yards  ? 

17.  How  many  cubic  feet  of  wood  are  there  in 
2  sticks  of  timber  each  of  which  is  18  feet  long, 
15  inches  wide,  and  10  inches  thick  ? 

18.  The  quantity  of  wood  contained  in  a  pile 
4  feet  high,  4  feet  wide,  and  1  foot  long  is  called 
1  CORD  FOOT  of  wood.     How  many  cubic  feet  are 
there  in  a  cord  foot?     How  many  cord  feet  are 
there  in  a  cord  ? 

19.  How  many  cubic  feet  of  wood  are  there  in 
a  pile  4  feet  wide,  6  feet  high,  and  24  feet  long  ? 
how  many  cord  feet  ?  how  many  cords  ? 

Before  working  out  each  of  the  next  four  examples,  decide 
whether  it  will  be  shorter  to  call  the  inches  fractions  of  a  foot  or 
to  reduce  them  to  decimals  of  a  foot.  If  you  use  decimals  carry 
out  the  result  to  the  nearest  hundredth  only. 

20.  How  many  cubic  feet  are  there  in  a  pile  of 
wood  4  ft.  2  in.  wide,  3  ft.  8  in.  high,  and  13  ft, 
long  ? 


B.]  Solid  Measure.  219 

21.  How  many  cords  are  there  in  a  pile  of  wood 
24  ft.  9  in.  long,  4  ft.  wide,  and  6  ft.  high  ? 

22.  How  many  cord  feet  are  there  in  a  load  of 
wood  8  ft.  long,  4  ft.  wide,  and  3  ft.  9  in.  high? 

Wood  prepared  for  the  market  is  generally  4  feet  long-,  and  a 
load  in  a  wagon  generally  contains  two  lengths  or  8  feet  in  length. 
If  a  load  is  8  feet  long,  4  feet  high,  and  4  feet  wide  it  contains  a 
cord. 

23.  How  many  cord  feet  are  there  in  a  load  of 
wood  8  ft.  long,  3  ft.  4  in.  wide,  and  2  ft.  6  in. 
high? 

24.  At  $8  a  cord,  what  is  the  cost  of  a  load  of 
wood  12  ft.  long,  4  ft.  wide,  and  3^  ft.  high  ? 

25.  At  $9  a  cord,  what  is  the  cost  of  a  load  of 
wood  8  ft.  long,  4  ft.  2  in.  wide,  and  5  ft.  4  in. 
high? 

26.  If  a  load  of  wood  is  8  ft.  long  and  3  ft.  4 
in.  wide,  how  high  must  it  be  in  order  to  contain 
a  cord  ? 

27.  What  is  the  cost  of  a  pile  of  wood  16  ft. 
long,  7^  ft.  high,  and  4  ft.  wide,  at  17.50  a  cord  ? 

28.  How  many  square  inches  are  there  in  the 
surface  of  a  cube,  each  of  whose  edges  is  2  inches  ? 

29.  How  many  square  yards  are  there  in  the 
surface  of  a  block  that  is  6  ft.  long,  4^  ft.  wide, 
and  3  ft.  high  ? 

The  next  31  examples  have  been  selected  with  the  permission 
of  the  publisher,  Seymour  Eaton,  Boston,  from  Practical  Men- 
suration, by  W.  V.  Wright. 

30.  How  many  shovelfuls  of  earth,  each   con- 
taining 16  cubic  inches,  are  there  in  a  cubic  yard  ? 


220  Square  and  Solid  Measures.         [§  8. 

31.  How  many  cubic  yards  of  earth  will  be  re- 
moved by  digging  a  cellar  12  feet  long,  10  feet 
wide,  and  5f  feet  deep? 

32.  How  many  cubic  feet  of  air  will  a  room  14 
feet  long,  12  feet  wide,  and  10  feet  high,  contain  ? 

33.  What  will  it  cost  to  dig  a  cellar  16  feet 
long,  12  feet  wide,  and  3£  feet  deep,  at  75  cents 
per  cubic  yard  ? 

34.  How  many   stones,  each   10   inches   by  8 
inches  by  6  inches,  will  it  take  to  build  a  wall  40 
rods  long,  6  feet  high,  and  2£  feet  thick? 

35.  What  weight  of  water  will  a  tank  contain, 
the  length  of  which  is  8  feet,  the  breadth  5J  feet, 
and  the  depth  7  feet?     [A  cubic  foot  of  water 
weighs  62 1  pounds.] 

36.  How  many  loads  of  gravel  will  be  required 
for  a  road  3  miles  long,  if  it  is  spread.  9  feet  wide 
and  8  inches  deep.   [A  cubic  yard  is  called  a  load.] 

37.  What  is  the  value  of  a  pile  of  wood  120 
feet  long,  16  feet  wide,  and  8  feet  high,  at  $4.50 
per  cord  ? 

38.  A  sled,  upon  which  four-foot  wood  is  piled 
cross- wise,  is  12  feet  long ;  how  high  should  the 
wood  be  piled  to  make  1 J  cords  ? 

39.  How  many  cords  of  wood  can  be  piled  into 
a  woodshed  24  feet  long,  16  feet  wide,  and  10  feet 
high? 

40.  A  wagon  is  12  feet  long  and  3^  feet  wide  ; 
how  high  must  wood  be  built  upon  it  to  make  1| 
cords  ? 

41.  How  many  cubic  feet  will  100  bushels  of 


B.]  Solid  Measure.  221 

grain  occupy?     [A  bushel  is  equal  to  nearly  1^ 
cubic  feet.] 

42.  How  many  bushels  of  oats  will  a  bin  8  feet 
by  6  feet  by  5  feet  contain  ? 

43.  How  many  cubic  feet  of  space  will  be  nec- 
essary in  which  to  store  600  bushels  of  corn  ? 

44.  A  bin  is  8  feet  by  10  feet ;  how  deep  will 
the  wheat  be  in  it  when  it  contains  400  bushels  ? 

45.  How  many  bushels  of  shelled  corn  will  a 
wagon-box  contain  that  is  12  feet  long,  3  feet  wide, 
and  20  inches  deep  ? 

46.  How  many  bushels  of  potatoes  can  be  stored 
in  a  bin  20  feet  by  8  feet,  and  6  feet  deep  ?     [A 
bushel   heaped  measure  contains  about  1^  cubic 
feet.] 

47.  How  deep  must  the  potatoes  be  in  a  wagon- 
box  10  feet  long  and  3  feet  wide  to  contain  20 
bushels  ? 

48.  How  many  bushels  of  turnips  will  a  wagon- 
box  contain  that  is  10  feet  long,  3  feet  wide,  and 
22  inches  deep  ? 

49.  If  a  gallon  contains  230  inches,  how  deep 
is  the   oil  in  a  tank  2  feet  square  to  contain  60 
gallons  ? 

50.  A  school-room  28  feet  by  30  feet,  and  12 
feet  high,  seats  72  children  ;  how  many  cubic  feet 
of  air  are  there  for  each  child  ? 

51.  How  high  should  the  ceiling  of  a  school- 
room which  is  28  feet  by  32  feet  be  to  allow  150 
cubic  feet  of  air  to  each  of  50  pupils  ? 

52.  The  ceiling  of  a  school-room  is  20  feet  high  ; 


222  Square  and  Solid  Measures.          [§  8. 

how  many  square  feet  of  floor  will  each  of  60  pupils 
have,  allowing  each  pupil  100  cubic  feet  of  air  ? 

53.  A  tank  20  yards  long  and  14  yards  broad 
will  hold  1,680  cubic   yards   of  water ;   find   its 
depth. 

54.  A  school-room  in  which  there  are  48  pupils 
and  one  teacher  is  35  feet  by   28  feet,  and  the 
ceiling  is  12  feet  high  ;  making  no  allowance  for 
furniture,  how  many  cubic  feet  of  air  will  the  room 
allow  for  each  person? 

55.  If  a  barrel  contains  4£  cubic  feet,  how  many 
barrels  of  lime  are  there  in  a  pile  30  feet  long,  22 
feet  wide,  and  2  feet  deep  ? 

56.  Find  the  cost  of  making  a  road  108  yards 
in  length  and  20  feet  wide  ;  the  soil  being  first  ex- 
cavated to  the  depth  of  1  foot  at  a  cost  of  25  cents 
per  yard  ;  rubble  then  being  laid  8  inches  deep  at 
20  cents  per  cubic  yard,  and  gravel  placed  on  the 
top  9  inches  thick  at  45  cents  per  cubic  yard. 

57.  If  a  ton  of  coal  occupies  40  cubic  feet,  what 
will  it  cost  to  fill  a  bin  12  feet  long,  6  feet  wide, 
and  5  feet  deep,  with  coal  at  $6.50  a  ton  ? 

58.  How  high  must  wood  be  piled  on  a  car  36 
feet  long  and  8  feet  wide  to  make  16  cords  ? 

59.  A  reservoir  is  24  feet  8  inches  long  by  12 
feet  9  inches  wide  ;  find  how  many  cubic  feet  of 
water  must  be  drawn  off  to  make  the  surface  sink 
1  foot. 

60.  Find  the  weight  of  10  boards,  each  30  feet 
long,  1  foot  wide,  and  1  inch  thick,  if  sawed  from 
wood,  one  cubic  foot  of  which  weighs  20  pounds. 


C.]  Board  Measure.  223 

C.  Board  Measure. 

A  board  1  foot  long,  1  foot  wide,  and  1  inch 
thick,  or  its  equivalent,  is  called  a  BOARD  FOOT  or 
1  FOOT  BOARD  MEASURE.  A  like  board  2  inches 
thick  contains,  therefore,  2  board  feet ;  if  2|  inches 
thick  it  contains  2|  board  feet ;  if  3  J  inches  thick 
it  contains  3|  board  feet,  etc. 

Boards  less  than  1  inch  thick  are  regarded  in 
measuring  as  inch  boards. 

1.  How  many  board  feet  are  there  in  a  board 
8  feet  long,  2  feet  wide,  and  1  inch  thick  ?  how 
many  in  a  board  8  feet  long,  2  feet  wide,  and  2| 
inches  thick  ? 

2.  How  many  board  feet  are  there  in  3  boards 
each  of  which  is  16  feet  long,  1^  feet  wide,  and  1 
inch  thick  ? 

3.  How  many  feet  (board  measure)  of  boards 
1  inch  thick  would  be  needed  to  lay  a  floor  18  ft. 
by  16  ft.  ? 

4.  At  $22  per  thousand  feet  find  the  cost  of 
3  boards,  each  12  ft.  by  2  ft.  and  1  in.  thick ; 

6  boards,  each  16  ft.  by  1|  ft.  and  1^  in.  thick ; 
8  boards,  each  15  ft.  by  2|-  ft.  and  \\  in.  thick. 
NOTE.     A  board  over  1J  or  2  inches  thick  is 
usually  called  a  plank. 

5.  At  f 24  per  thousand  feet  find  the  cost  of 

2  planks,  each  14  ft.  by  2  ft.  and  2*  in.  thick ; 

3  planks,  each  8  ft.  by  1|-  ft.  and  3  in.  thick ; 

4  planks,  each  6  ft.  by  2  ft.  and  3*  in.  thick. 

6.  How  many  board  feet  are  there  in  a  stick  of 


224  Square  and  Solid  Measures.          [§  8. 

timber  20  feet  long,  18  inches  wide,  and  18  inches 
thick  ? 

7.  At  $  18  per  thousand  feet,  what  is  the  cost  of 
enough  inch  boards  for  a  fence  6  feet  high  and  230 
feet  long?     At  $19  per  thousand  feet,  what  is  the 
cost  of  enough  lumber  for  3  cross  pieces,  each  2 
inches  thick  and  4  inches  wide,  to  run  the  entire 
length  of  the  fence? 

8.  At  $20  per  thousand  feet,  what  is  the  cost  of 
40  scantlings  (narrow  planks)  each  16  feet  long, 
4  inches  wide,  and  3  inches  thick  ? 

9.  A  board  walk  is  to  be  made  of  boards  6 
inches  wide  and  1  inch  thick,  and  of  cross  pieces 
4  inches  wide  and  2  inches  thick.     The  walk  is  to 
be  1056  feet  long,  8  boards  wide,  with  cracks  or 
spaces  between  them,  amounting  in  all  to  2  inches 
in  width   [this  will  make  the  entire  width  8x6 
in. +  2  in.  =  50  in.],  and  the  cross  pieces,  to  which 
the  boards  are  to  be  nailed,  are  to  be  2  ft.  8  in. 
apart.    Find  the  cost  of  the  lumber  needed  for  the 
walk  if  the  boards  cost  $18  per  thousand  feet  and 
the  cross  pieces  $19  per  thousand  feet. 

10.  How  many  feet  of  boards  1^  inches  thick 
would  be  required  for  a  coal  bin,  open  at  the  top, 
8  feet  high,  7  feet  long,  and  6  feet  wide,  if  built  in 
the  corner  of  a  cellar  so  that  the  cellar  walls  may 
form  two  sides  of  the  bin  and  the  cellar  floor  may 
form  the  bottom  ? 

11.  How  many  board  feet  are  there  in  a  plank 
3  inches  thick,  16  feet  long,  18  inches  wide  at  one 
end  and  14  inches  wide  at  the  other  end  ?     [The 
average  width  is  |  of  18  +  14  inches,  or  1^  feet.] 


C.]  Board  Measure.  225 

12.  How  many  feet  board  measure  are  there  in 
a  board  16  feet  long,  9  inches  wide,  and  J  inch 
thick  ?     Ans.  12  ft. 

13.  How  many  feet  of  boards  |  of  an  inch  thick 
would  be  required  for  the  top  of  a  dining -table 
12  ft.  by  4  ft.  ? 


SECTION  IX. 
DIVISORS,   FACTORS,   AND  MULTIPLES. 

[The  portions  of  this  section  inclosed  in  starred  brackets  *[  ]* 
may  be  omitted  by  pupils  whose  time  is  limited.] 

IN  our  study  of  fractions  we  have  learned  that 
the  value  of  a  fraction  is  not  altered  by  dividing 
both  the  numerator  and  the  denominator  by  the 
same  number.  An  application  of  this  principle 
to  the  reduction  of  fractions  to  their  lowest  terms 
leads  us  to  seek  for  a  method  of  finding  the  divi- 
sors of  numbers.  Let  us  begin  with  numbers  so 
small  that  the  divisors  can  be  found  more  easily 
by  inspection  than  by  the  aid  of  any  other  method, 
and  proceed  gradually  to  larger  numbers. 

1.  What  divisors  has  15  ?     Ans.  3  and  5.* 

2.  What  divisors  has  each  of  the  following  num- 
bers? 

6,  10,  14,  21,  22,  25,  26,  33,  34,  35. 

3.  Of  the  numbers  just  mentioned  which  have 
2  as  a  divisor  ? 

*  Strictly  speaking,  1  is  a  divisor  of  15,  because  1  is  contained 
in  it  15  times  with  no  remainder  ;  and  15  is  a  divisor  of  15  with 
a  quotient  1.  It  is  not  customary,  however,  except  in  special 
cases,  to  regard  1  and  the  number  itself  as  divisors  of  a  number. 


Divisors.  227 

NOTE.     Numbers  divisible  by  2  are  called  EVEN 
NUMBERS,  and  numbers  not  divisible  by  2  are  called 

ODD   NUMBERS. 

4.  Give  all  the  even  numbers  from  2  to  50  in- 
clusive. 

5.  Give  all  the  odd  numbers  from  1  to  49  in- 
clusive. 

6.  2  is  a  divisor  of  10  and  therefore  of  twice  10, 
three  times  10,  or  of  any  number  of  tens.     Is  2  a 
divisor  of  66?     Ans.  66  =  60  +  6;    2  is  a  divisor 
of  60  and  of  6,  and  therefore  of  their  sum. 

7.  Is  2  a  divisor  of  46  ?  33  ?  50  ?  128  ?  39  ? 

8.  Which  of  the  numbers  20,  25,  30,  50,  65, 
100,  135,  and  150  contain  10  ?  which  contain  5  ? 

9.  What  must  the  last  figure  of  a  number  be 
in  order  that  it  may  contain  10  ?  5  ? 

10.  Is  4  a  divisor  of  100  ?  of  300  ?  of  700  ?  of 
1100  ?  of  1400  ?  of  2100  ?  of  2300  ? 

11.  Is  4  a  divisor  of  1112?    [Suggestion:  1112 


12.  State  whether  4  is  a  divisor  of  1316,  of  2984, 
of  1981,  of  117,644,  of  1060. 

NOTE.  From  what  precedes  we  conclude  that 
any  number  contains  4  if  the  number  formed  by 
its  last  two  figures  contains  4,  or  if  its  last  two 
figures  are  zeros. 

13.  Is  8  a  divisor  of  100?  of  1000?  of  2000? 
of    16000?  of   16168   [16000  +  168]?    of    21248 
[21000  +  248]?  of  26001  [26000  +  1]?  of  169421 
[169000  +  421]? 

NOTE.     From  the  last  example  we  may  learn 


228        Divisors,  Factors,  and  Multiples.      [§  9. 

that  any  number  contains  8  if  the  number  formed 
by  its  last  three  figures  contains  8  or  if  its  last 
three  figures  are  zeros. 

*[If  we  divide  10  by  9  we  get  a  remainder  1. 
Thus  10-9  +  1 

therefore  20-2x94-2 

40-4x9  +  4 
60  =  6x9  +  6,  etc. 

From  this  we  see  that  when  we  divide  any  num- 
ber of  tens  (less  than  9  tens)  by  9  the  remainder 
is  always  the  same  as  the  number  of  tens. 

If  we  divide  100  by  9  we  get  11  times  9  with  a 
remainder  1. 

Thus  100-99  +  1 

therefore  300-3x99  +  3 

700-7x99  +  7,  etc. 

From  this  we  see  that  when  we  divide  any  num- 
ber of  hundreds  (less  than  9  hundreds)  by  9,  the 
remainder  is  always  the  same  as  the  number  of 
hundreds. 

Let  us  see  if  765  contains  9. 
700-7x99  +  7 
*60-6x    9  +  6 
5-Ox    9+5 
Therefore  765-7x99  +  6  x9+[7  +  6  +  5]. 

We  know  that  7  x  99  and  6x9  each  contains  9, 
and  also  that  the  sum  of  the  remainders  [7  +  6  +  5 
or  18]  contains  9  ;  therefore  the  sum  of  all  these 
quantities,  or  the  number  itself,  contains  9.  But 
the  remainders  7,  6,  and  5  are  the  figures  of  the 


Divisors.  229 

number  765  ;  we  may  say,  then,  that  765  contains 
9  because  the  sum  of  its  figures  contains  9.]* 

In  like  manner  we  may  show  that  any  number 
contains  9  if  the  sum  of  its  figures  contains  9. 

14.  In  which  of  the  following  numbers  is  9  con- 
tained without  a  remainder  ? 

279,  842,  1980,  2223,  1116738. 

*[15.  Does  762  contain  3? 

Separating  this  number  into  parts,  as  was  done 

before,  we  get 

700-7x99  +  7 
60  =  6x    9  +  6 
2=  2 

Therefore  762  =  7  x  99  +  6  x  9+  [7  +  6  +  2]. 

Here  each  of  the  first  two  parts  (7  x  99  and 
6x9)  contains  9  and  therefore  3  ;  the  third  part 
(the  sum  of  the  remainders  7,  6,  and  2)  also  con- 
tains 3.  Therefore  the  number  762  contains  3.]* 

In  like  manner  we  may  show  that  any  number 
contains  3  if  the  sum  of  its  figures  contains  3. 

16.  Which  of  the  numbers  18,  31,  72,  129,  684, 
1728,  1333,  2168,  2121,  9843  contain  3? 

NOTE.  To  know  whether  a  number  contains 
7,  11,  13,  17,  etc.,  we  must  actually  try  to  divide, 
in  order  to  learn  whether  we  can  do  so  without  a 
remainder. 

17.  Find  by  trial  whether  either  of  the  numbers 
7,  11,  13,  or  17  is  contained  in  9680  ;  in  5780 ;  in 
1690  ;  in  3367  ;  in  18432. 


230         Divisors,  Factors,  and  Multiples.      [§  9. 

18.  Which  of  the  following  numbers  has  divi- 
sors (other  than  itself  and  1)  ? 

1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11, 12, 13, 14, 15, 16, 
17,  18,  19,  20. 

19.  Of  the  numbers  just  given,  which  have  no 
divisors  (except  themselves  and  1)  ? 

NOTE.     A  number  which  has  no  divisors  except 
itself  and  1  is  called  a  PRIME  NUMBER. 

20.  Name  all  the  prime  numbers  between  20 
and  50. 

21.  Name  all  the  prime  numbers  between  50 
and  76. 

NOTE.     A  number  which  has  other  divisors  than 
itself  and  1  is  called  a  COMPOSITE  NUMBER. 

22.  Name  all  the  composite  numbers  from  20  to 
50  inclusive. 

23.  Name  all  the  composite  numbers  from  50 
to  75  inclusive. 

24.  What  is  the  smallest  prime  number? 

25.  What  even  number  is  also  a  prime  number? 

26.  What  are  the  prime  divisors  (those  divisors 
which  are  prime  numbers)  of  12  ?   Ans.  2,  2,  and  3. 

27.  What  are  the  prime  divisors  of  4?  of  6? 
of  9?  of  21? 

28.  Find  (a)  the  prime  divisors  of  30,  (6)  the 
other  divisors  of  30. 

(a)  First  dividing  by  2,  the  smallest 


prime  divisor,  we  get  15. 

Now,  since  any  divisor  of  15  is  also  a 
divisor  of  twice  15  or  30,  we  will  pro- 
ceed in  the  same  way  with  15. 


SO 
15 


Divisors.  231 

Dividing  15  by  its  smallest  prime  divisor  (3) 
we  get  5,  and  dividing  5  by  5  we  get  1. 

The  prime  divisors  of  30  thus  obtained  are  2,  3, 
anjd  5.  The  work  may  be  briefly  arranged  as 
shown  in  the  margin  of  the  preceding  page. 

By  reversing  the  work  just  done  we  get  5x3x2 
=  30,  or  the  product  of  the  prime  divisors  of  a 
number  is  equal  to  the  number  itself. 

b.  The  other  divisors  of  30  are  6  (3  x  2),  10 
(2x5),  and  15  (3x5). 

29.  Find  the  prime  divisors  of  28.     Ans.  2,  2, 
and  7.     What  other  divisors  has  28  ?    Ans.  4  and 
14.     What  is  the  product  of  the  prime  divisors  ? 

30.  Find  the  prime  divisors  of  330. 

31.  Find  the  prime  divisors  of  88.     What  other 
divisors  has  88  ?     What  is  the  continued  product 
of  the  prime  divisors  ? 

32.  Find  the  prime  divisors  of  each  of  the  fol- 
lowing numbers:  15, 18,  20,  24,  28,  42,  48,  72,  98. 

33.  Find  the  prime  divisors  of  each  of  the  fol- 
lowing numbers :  112,  114,  120,  387,  432,  846. 

34.  Find  the  prime  divisors  of  2064.     Ans.  2, 

2,  2,  2,  3,  43. 

35.  Find  the  prime  divisors  of  2480. 

36.  Find  the  prime  divisors  of  6006.     Ans.  2, 

3,  7,  11,  13. 

37.  Find  the  prime  divisors  of  2093. 

38.  Which  of  the  numbers  324,  174,  222,  1728 
contain  6  ? 

39.  Show  that  an  even  number  which  contains  3 
will  also  contain  6. 


232         Divisors,  Factors,  and  Multiples.      [§  9. 

40.  Which  of  the  numbers  198,  594, 1278, 1638 
contain  18? 

41.  Show  that  an  even  number  which  contains 
9  will  also  contain  18. 

42.  Show  that  a  number  ending  with  5  or  0 
which  contains  3  will  also  contain  15. 

43.  Which  of  the  numbers  495,  585,  765,  360 
contain  45  ? 

44.  Find  all  the  divisors  common  to  30  and  42. 
Ans.  2,  3,  and  6. 

45.  Find  all  the  divisors  common  to  30,  42,  and 
52.     Ans.  2. 

NOTE.  When  a  number  will  divide  two  or  more 
numbers  at  the  same  time  it  is  said  to  be  a  COM- 
MON DIVISOR  of  those  numbers,  and  the  greatest 
number  that  will  divide  two  or  more  numbers  at 
the  same  time  is  called  their  GREATEST  COMMON 
DIVISOR  (g.  c.  r/.). 

46.  What  is  the  g.  c.  d,  of  30  and  42?   Ans.  6. 

47.  What  is  the  g.  c.  d.  of  30,  42,  and  52? 
Ans.  2. 

48.  Find  the  g.  c.  d.  of  231  and  273. 

231-3x7x11 
273-3x7x13 

The  only  common  divisors  are  3  and  7.  There- 
fore the  g.  c.  d.  is  3  x  7,  or  21. 

49.  Find  the  g.  c.  d.  of  16  and  36. 

50.  Find  the  g.  c.  d.  of  18  and  42. 

51.  Find  the  g.  c.  d.  of  21  and  56. 

52.  Find  the  g.  c.  d.  of  56  and  264. 

53.  Find  the  g.  c.  d.  of  123  and  642. 


Divisors. 


233 


54.  Find  the  g.  c.  d.  of  32,  96,  and  1432. 

55.  Find  the  g.  c.  d.  of  108,  45,  18,  and  63. 

56.  Find  the  g.  c.  d.  of  8  and  24. 

57.  Find  the  g.  c.  d.  of  18,  36,  12,  48,  and  42. 

58.  Find  the  g.  c.  d.  of  114,  102,  78,  and  66. 

59.  Reduce  |-|-J  to  its  lowest  terms. 

We  may  reduce  to  lower  terms,  step  by  step, 
until  we  get  the  lowest  terms,  thus  : 


231 
273 


-11 
=  9l' 


77 
91 


11 
13 


281 

273 


11 
13' 


Or  we  may  first  find  the  g.  c.  d.  of  231  and  273. 
and  then  divide  both  terms  by  it,  thus  : 


60.  Reduce 

61.  Reduce 

62.  Reduce 

63.  Reduce 

64.  Reduce 


f  $ 


- 
"13^ 

to  its  lowest  terms. 
to  its  lowest  terms. 
to  its  lowest  terms. 
Q  to  its  lowest  terms. 
-  to  its  lowest  terms. 


*[In  some  cases  the  greatest  common  divisor  of 
two  or  more  numbers  cannot  readily  be  found  by 
the  method  indicated  above.  We  will,  therefore, 
proceed  to  study  some  of  the  properties  of  the 
greatest  common  divisor  and  to  deduce  from  them 
a  method  that  we  can  readily  use  in  all  cases. 

I.  A  common  divisor  of  any  two  numbers  is  a 
divisor  (a)  of  their  sum,  and  (&)  of  their  differ- 
ence. 

(a)  Take  64  and  152,  for  example.  If  8  is  a 
common  divisor,  then  each  is  an  exact  number  of 


234         Divisors,  Factors,  and  Multiples.      [§  9. 

8's  ;  therefore  their  sum  must  be  an  exact  number 
of  8's,  i.  e.,  it  must  contain  8  as  a  divisor. 

(6)  If  each  is  an  exact  number  of  8's,  then  the 
larger  must  contain  more  8's  than  the  smaller,  and 
their  difference,  which  shows  how  many  more  8's 
there  are  in  one  than  in  the  other,  must  contain  8 
as  a  divisor. 

II.  The  g.  c.  d.  of  two  numbers  is  the  greatest 
common  divisor  of  the  smaller  and  of  the  remain- 
der which  we  get  after  dividing  the  larger  by  the 
smaller.  Take  64  and  152,  for  example,  of  which 
8  is  the  g.  c.  d.  Since  each  is  an  exact  number  of 
8's,  twice  or  three  times,  or  four  times,  or  any  num- 
ber of  times  either,  will  be  an  exact  number  of  8's. 
Now,  when  we  divide  the  larger  by  the  smaller, 
we  subtract  from  the  larger  as  many  times  the 
smaller  as  possible.  In  this  case  we 
subtract  128  or  two  times  the  smaller, 
and  get  for  a  remainder  24,  a  number 
which  shows  by  how  many  8's  the  larger 
exceeds  two  times  the  smaller ;  this  remainder,  then, 
contains  8  (the  g.  c.  d.  of  the  two  numbers  that  we 
started  with),  and  we  may  say  that  the  g.  c.  d.  of 
the  two  numbers  is  a  common  divisor  of  either  — 
the  smaller,  if  we  please  —  and  this  remainder. 
The  smaller  number  and  the  remainder  have  no 
common  divisor  larger  than  8,  because  any  num- 
ber that  divides  the  remainder  and  the  smaller  will 
also  divide  the  larger,*  and  8  (we  were  told  at 
the  start)  is  the  largest  number  that  will  divide 

*  A  divisor  of  64  and  24  is  also  a  divisor  of  2  X  64  +  24. 


Divisors.  235 

both  numbers.  The  g.  c.  d.  of  the  smaller  and  the 
remainder,  then,  is  the  same  as  that  of  the  two 
numbers. 

III.  Now,  by  the  principle  just  explained,  we 
know  that  the  g.  c.  d.  of  24  and  64  is  also  the  g.  c. 
d.  of   the  next  remainder  (8)  and 

the  smaller  number  (16)  ;  but  since  i^ 

8    goes   in    16  twice,  with   nothing  ~Q\Tftr<> 

over,  it  is  itself  the  g.  c.  d.  of  8  and 

16,  and  therefore  of  the  numbers  we  ~Q~ 

started  with. 

IV.  We  may  now  say  that  to  get  the  g.  c.  d.  of 
two  numbers,  we  may  divide  the  larger  by  the 
smaller,  and  then  the  smaller  by  the  remainder •, 
if  there  be  any,  and  then  continue  dividing  the 
last  divisor  by  the  last  remainder  until  nothing 
remains.     TJie  last  divisor  will  be  the  g.  c.  d. 

Find  the  g.  c.  d.  of  22176  and  23328. 
22176)23328(1 
22176 

1152)22176(19 
1152 
10656 
10368 


288)1152(4 
Ans.  288.  1152 

65.  Find  the  g.  c.  d.  of  11385  and  16335. 

66.  Reduce  -£££*  to  its  lowest  terms. 

V  I  2  V 

67.  Reduce  $$$$  to  its  lowest  terms. 

68.  Reduce  to  its  lowest  terms 


236  Divisors,  Factors,  and  Multiples.      [§  9. 

69.  Reduce  to  its  lowest  terms  §4M« 

o  1  J.  o 

70.  Reduce  to  its  lowest  terms  |r;;|^. 

71.  Reduce  to  its  lowest  terms  \\^r 

72.  Reduce  to  its  lowest  terms  ffify* 

73.  Reduce  to  its  lowest  terms  JjiilJ". 

74.  Reduce  to  its  lowest  terms  [: 

75.  Reduce  to  its  lowest  terms  ^§||.]* 

76.  a.  Add  together  \,  -J-,  and  \. 


6.  Subtract  \  from  \.    Ans.  \  -  '  = 


. 
1.Z 

NOTE.  In  examples  like  76  a  and  76  6  we  have 
to  reduce  fractions  to  a  common  denominator.  Any 
number  that  will  contain  each  of  the  denominators 
will  serve  for  a  common  denominator,  but  the 
smallest  number  that  will  do  this  is,  of  course, 
the  easiest  to  manage. 

The  next  few  examples  lead  up  to  a  method  of 
finding  the  smallest  number  that  will  contain  two 
or  more  different  numbers  as  divisors. 

77.  a.  15  is  the  product  of  what  two  numbers  ? 

Ans.  3  and  5. 
b.  18  is  the  product  of  what  numbers  ? 

A?IK.  2  and  9,  or  2,  3,  and  3.    . 
NOTE.     Those  numbers  which  when  multiplied 
together  will  produce  a  given  number  are  called 
its  FACTORS. 

78.  What  are  the  factors  of  30  ? 

79.  What  are  the  factors  of  39  ? 


Factors  and  Multiples.  237 

80.  What  are  the  prime  factors  of  154  ? 
NOTE.     From  what    precedes  we  see  that   the 

factors  of  a  number  are  also  divisors  of  the  num- 
ber, and  that  the  prime  factors  are  the  same  as  the 
prime  divisors. 

81.  What  is  the  number  whose  factors  are  3 
and  5  ?     Ans.  15. 

82.  What  is  the  number  whose  factors  are  2,  7, 
and  11  ? 

83.  What  is  the  number  whose  factors  are  2,  2, 
3,  3,  and  3  ? 

NOTE.  Any  number  which  contains  another 
number  as  a  factor  or  divisor  is  called  a  MULTIPLE 
of  that  other  number.  Thus  15  is  a  multiple  of  3 
and  also  of  5  ;  18  is  a  multiple  of  2  and  also  of  3 
and  also  of  9. 

84.  Of  what  numbers  is  30  a  multiple  ? 

85.  Of  what  numbers  is  48  a  multiple  ? 

86.  What  is  the  least  number  which  is  a  multi- 
ple of  5  and  at  the  same  time  of  3  ? 

Ans.  5  x  3,  or  15. 

87.  What  is  the  least  number  that  is  a  multiple 
of  3,  5,  and  2  ?     Ans.  3  x  5  x  2,  or  30. 

NOTE.  The  least  number  that  is  a  multiple  of 
two  or  more  numbers  is  called  their  LEAST  COM- 
MON MULTIPLE  (I.  c.  m.). 

88.  What  is  the  I.  c.  m.  of  15  and  21  ? 
Solution :  Separating  each  into  its  prime  fac- 

15  =  3  x  5 
tors,  we  get  91  _  o  x  7-    In  order  to  contain  15,  the 

required  number  must  contain  3  and  5  ;  and  in  or- 


238         Divisors,  Factors,  and  Multiples.      [§  9. 

tier  to  contain  21  it  must  contain  3  and  7.  That 
number,  then,  which  contains  only  3,  5,  and  7 
must  be  the  required  number  ;  therefore  3x5x7  = 
105  is  the  I.  c.  m. 

89.  a.  What  is  the  I.  c.  m.  of  18  and  21  ? 
Solution :    Separating    each    number    into   its 

r  18  =  2x3x3 

prime  factors,  we  get    91-3x7 

We  see  from  the  above  that  the  least  number 
which  will  contain  18  and  21  must  contain  2,  3 
twice,  and  7  ;  therefore  the  1.  c.  m.  =  2x3x3x7  = 
126. 

b.  What  is  the  L  c.  m.  of  36,  48,  and  60  ? 

Ans.  720. 

90.  What  is  the  L  c.  m.  of  6,  20,  and  45  ? 
Solution :  Separating  each  number  into  its  prime 

factors  we  get : 

6=2x3        ) 

20  =  2x2x5  [•'•  the/.  c-  ™- 
45  =  3x3x5)      -2x2x3x3x5  =  180. 

NOTE.  From  the  preceding  illustrations  we  see 
that  the  L  c.  m.  of  two  or  more  numbers  is  the 
product  of  all  the  different  prime  factors  of  all 
the  numbers,  each  prime  factor  being  used  the 
greatest  number  of  times  that  it  occurs  in  any 
one  of  the  numbers. 

In  the  last  example,  for  instance,  2  occurs  twice 
as  a  factor  in  20,  and  3  occurs  twice  in  45  :  the 
only  remaining  prime  factor  is  5,  which  occurs 
only  once  in  the  same  number ;  therefore  the  L  c.  m. 
=  2x2x3x3x5  =  180. 


Factors  and  Multiples. 


239 


The  following  is  another  method  of  finding  the 
L  c.  m.,  although  the  principle  is  the  same. 
6     20     45 


3     10     45 


1     10     15 


2      3 
L  c.  m.  =  2x  3x5x2x3  =  180. 

This  method  consists  only  in  a  briefer  arrange- 
ment of  the  following  work  which  we  should  do  in 
finding  the  prime  factors  of  each  number  sepa- 
rately : 


2|6_ 
3 


20 


45 
15 


6  =  2x3       20  =  2x5x2       45  =  3x5x3 
91.  Find  the  7.  c.  m.  of  21,  33,  and  28. 


21     33 


7     11 


28 
"28 

~  4 


1     11       4    Ans.  3x7x11x4. 
92.  Find  the  L  c.  m.  of  100,  400,  and  1000. 

In  this  case  the  number 


100 


100     400 


JLOOO 

"To" 


100  is  a  common  divisor 
of  all  the  numbers,  we 
can  therefore  just  as  well 
divide  by  it  at  once  as 
divide  by  its  prime  factors  one  after  the  other. 

93.  Find  the  L  c.  m.  of  8  and  12. 

94.  Find  the  L  c.  m.  of  8  and  14. 

95.  Find  the  I.  c.  m.  of  9  and  15. 


240         Divisors,  Factors,  and  Multiples.     [§  9. 

96.  Find  the  /.  c.  m.  of  15  and  18. 

97.  Find  the  I.  c.  m.  of  10,  14,  and  15. 

98.  Find  the  L  c.  m.  of  15,  24,  and  35. 

99.  Find  the  L  c.  m.  of  30,  48,  and  56. 

100.  Find  the  L  c.  m.  of  32,  72,  and  120. 

101.  Find  the  /.  c.  m.  of  42,  60,  and  125. 

102.  Find  the  L  c.  m.  of  250,  180,  and  540. 

103.  Reduce  f  and  |  to  the  least  common  de- 
nominator and  then  add  them  together. 

104.  Reduce  |  and  T5?  to  the  least  common  de- 

^r  1  O 

nominator  and  then  add  them  together. 

105.  Reduce  f,  ^,  ^5,  and  ^  to  their  least 
common  denominator. 

106.  Find  the  greatest  common  divisor  of  48 
and  130. 

107.  Reduce  ^,  f ,  ^,  and  -J-J  to  their  least  com- 
mon denominator. 

108.  Subtract  15|  from  18f . 

109.  Reduce  |  and  -|  to  the  least  common  de- 
nominator and  then  add  them. 

110.  Reduce  f  and  T5?  to  the  least  common  de- 
nominator and  then  add  them. 

111.  Reduce  -f%  and  -fy  to   the   least   common 
denominator  and  then  add  them. 

112.  Reduce   ^   and  -fa  to  the  least  common 
denominator  and  then  add  them. 

113.  Reduce   fa   and  -^  to  the  least  common 
denominator  and  then  add  them. 

114.  Reduce  T|,  /y,  and  \\  to  the  least  common 
denominator  and  then  add  them. 

115.  Reduce  |,  f,  -f^,  and  ^y  to  the  least  com- 
mon denominator  and  then  add  them. 


Factors  and  Multiples.  241 

*[116.  a.  Find  the  I.  c.  ra.  of  407  and  481.  In 
a  case  of  this  kind,  where  none  of  the  prime  factors 
of  either  number  can  be  found  by  inspection,  it  is 
best  to  find  first  the  g.  c.  d.  In  this  example  we 
shall  find  the  g.  c.  d.  to  be  37,  which  is  contained  11 

times  in  407  and  13  times  in  481  .-. )  .    .  _. 

and  the  I.  c.  m.  is  11  x  13  x  37  =  5291, 
b.  Find  the  I.  c.  m.  of  731  and  817. 

117.  Find  the  I.  c.  m.  of  451  and  943. 

118.  Find  the  I.  c.  m.  of  217  and  341. 

119.  Find  the  I.  c.  m.  of  203  and  319.]* 

120.  Find  the  I.  c.  m.  of  17  and  31  and  2. 

121.  Find  the  I.  c.  m.  of  7,  13,  and  3. 

NOTE.  Where,  as  in  the  last  two  examples, 
two  or  more  numbers  have  no  common  factors, 
their  I.  c.  m.  is  evidently  their  product. 

122.  Find  the  I.  c.  m.  of  3,  13,  and  31. 

123.  What  is  the  I.  c.  m.  of  20,  24,  and  36  ? 

124.  Add  f,  |,  2T%,  and  8&. 

*[125.  What  is  the  g.  c.  d.  of  1181  and 
2741?]* 

126.  Reduce  |,  T%,  and  T7y  to  a  common  denom- 
inator. 

127.  Name  all  the  prime  numbers  in  the  series 
of  numbers  from  1  to  29  ;  resolve  all  the  composite 
numbers  into  their  prime  factors  ;  and  name  all 
the  perfect  squares. 

128.  Add  together  f ,  Jf,  and  T4^,  and  from  their 
sum  subtract  -. 


242         Divisors,  Factors,  and  Multiples. 

129.  a.  Find  the  g.  c.  d.  of  12,  30,  and  45. 
[The  g.  c.  d.  of  12  and  30  is  6  ;  and  the  g.  c.  d. 
of  6  and  45  is  3.     Therefore  the  answer  is  3.] 

6.  Find  the  g.  c.  d.  of  720,  336,  and  1736. 

130.  Reduce  £-J|  $-§•  to  its  lowest  terms. 

131.  Reduce   T^,  {£,  ^,  -fa,  and  ^  to  their 
least  common  denominator,  add  them  and  reduce 
the  sum  to  its  simplest  form. 

132.  Find  the  g.  c.  d.  and  the  I.  c.  m.  of  630, 
840,  and  2772. 

133.  Find  the  g.  c.  d.  and  /.  c.  m.  of  144  and 
780. 

134.  Reduce  J,  f,   j3^,  and  ||  to   their   least 
common  denominator. 

135.  Subtract  15J  from  18 J. 

136.  Reduce  £|f  $$  to  its  lowest  terms. 

137.  What  is  the  g.  c.  d.  of  the  two  numbers 
4760  and  3432  ? 

138.  What  is  the  I.  c.  m.  of  48,  98,  21,  and  27  ? 

139.  What  is  the  g.  c.  d.  of  1872  and  432  ? 
[Obtain  the  answer  by  factoring.] 

140.  Find  the  g.  c.  d.  of  187  and  153 ;  also 
their  I.  c.  m. 


SECTION  X. 
CANCELLATION  AND  ANALYSIS. 

1.  How  many  tons  of  coal  at  $6  a  ton  can  be 
bought  for  15  tons  of  hay  at  $18  a  ton? 

Solution :  15  tons  of  hay  at  118  a  ton  is  worth 
15  x  18  dollars,  or  $270.  As  many  tons  of  coal  at 
$6  a  ton  can  be  bought  for  $270  as  6  is  contained 
in  270,  or  45  tons. 

The  answer  just  found  was  got  by  dividing  15 
times  18  by  6.  We  may  say,  then,  that  the  an- 
swer is  equal  to  — ~ —  tons.  Now,  dividing  both 

the  dividend  and  the  divisor  first  by  3,  and  then 

5       9       . 

VM   ^    j'jjA 

by  2,  we  get  -  — - —  =  45,  and  thus  save  ourselves 

* 

some  time  and  labor.     Striking  out  the  common 
factors  from   the  divisor   and   dividend   is  called 

CANCELLATION. 

2.  If  6  men  can  build  a  stone  wall  40  ft.  long, 
5  ft.  high,  and  2  ft.  thick  in  6  days,  how  long  will 
it  take  them  to  build  a  wall  80  ft.  long,  5  ft.  high, 
and  3  ft.  thick? 


244  Cancellation  and  Analysis.          [§  10. 

Solution  : 

6  men  build  40  x  5  x  2  cu.  ft.  in  6  dys. 

,    .,,40x5x2 
o  men  build  -  ~  -  cu.  ft.  in  1  day. 

6  men,  to  build  80  x  5  x  3  cu.  ft.,  will  require 

* 

OA     „     0  .  40  x  5  x  2   ,  §0  x  £  x  3  x  6    _ 

80xox3-l--  -  days,  or  -  ---—   days, 

b  40  x  f  x  g 

or  18  days.     Ans. 

NOTE.  If,  in  the  solution  of  any  problem,  there 
is  to  be  a  series  of  multiplications  and  divisions  it 
is  always  well  first  to  express  them  all,  and  then 
to  cancel  if  possible. 

3.  Divide  $  of  f  of  2£  by  -J,. 

Solution  : 

First  expressing  the  successive  steps  and  then 

±31 

-----------  &,    .._  a~w   §    _    f    ---  2     -       O  I  ~  I  X  ?  X  2  *   if 


4.  Divide  1£  by  1  J.     Multiply  1J  by  1|. 

5.  Multiply  48  by  ^.     Divide  ^  by  ^. 

6.  Reduce  ^  —  ^-^      i  to  its  simplest  form. 


7.  From  |  of  f  take  J  of  f  . 

8.  Divide  ||  x  721  by  |  of  f  of  9|. 

91  4 

9.  Reduce  ^rl  to  a  simple  fraction.     Reduce 

Sf 

to  a  simple  fraction. 


Cancellation  and  Analysis.  245 

10.  What  is  the  product  of  |  of  ^  of  15  and 

15   Of  11  1? 

11.  Divide  100  by  4£. 

12.  Multiply  ||  by  ^  of  2J. 

43" 

13.  From  |  of  •£%  subtract  ^  of 

14.  Divide  &  of  ^  of  3J  by 

15.  Multiply  f  of  tf  of  41  by 

16.  From  fa  of  If  subtract  $  of 

17.  Divide  V-  of  ^  of  If  by    o 


18.  From  |-  of  f  f  subtract  ^  of  2J. 

19.  Divide  J  |  of  W  of  IS?  by  ^. 

^2        9 

20.  Divide  if  1  by  42. 

21.  Divide  |  of  ||  by  ^T  of  f  f. 

182 

22.  Reduce  ^  —  .  Q7  f  r  to  its  simplest  form 

I  of  I  of  I 

23.  Add  S,  1,  and  ^  of  f  . 

°$ 

24.  What  part  of  6  is  2  ? 

25.  What  part  of  T\  is  £  ? 

26.  What  part  of  f  is  f  ? 

27.  Divide  J  of  f  of  2  J  by  -. 


28.  Divide  TV  of  T^  of  81  by    -  . 

29.  From  3£  subtract  (^  of  -^  of  1|)  -r 


246  Cancellation  and  Analysis.          [§  10. 

30.  Subtract  J  of  f  from  f  of  -jf  ;  add  to  the 

«5 

remainder  -^g  ;  divide  the  result  by  6  j. 

31.  Add  -jj-|-  and  -=|  ;  divide  the  result  by  7£f  . 

"To  '  8 

3 

32.  From  \  of  If  take  -^-,  add  to  the  remain- 

^2" 

der  |,  and  divide  the  result  by  6f  . 

33.  From  ^  of  2|  subtract  the  product  of  0.075 
and  1^,  and  divide  the  remainder  by  12. 

34.  Divide  10  times  (£  of  -^  of  9&)  by  -||. 

35.  From  5    subtract  $&  +         of        of 


71  315 

36.  From  the  sum  of          and  --  subtract 


aud  divide  the  result  by  the  product  of  3^  and 

37.  Divide  (2}xJL)  by  (2J-lf). 

38.  Divide-        by  |  of  (-1^1  ). 


- 
39.  Add     to 


40.  Add        to 

41.  Add  W  of  -^  to  if. 

-••ii        19 

42.  Divide  0.75  by  -||x  0.081. 
•*o.    TY  iictt  10  tuc 

. 

44.  How  many  tiles  8  inches  square  will  cover 
a  hearth  12  ft.  wide  and  16  ft.  long? 


Cancellation  and  Analysis.  247 

45.  If  12  tailors  can  make  13  suits  of  clothes  in 
7  days,  how  many  tailors  will  it  take  to  make  the 
clothes  of  a  regiment  consisting  of  494  soldiers  in 
19  days. 

Solution  : 

12  tailors  make  13  suits  in  7  days. 

1  tailor  makes  -if  suits  in  7  days  (T^  as  many 
as  12  tailors). 

13 
1  tailor  makes  ^ — =,  suits  in  1  day. 

13x19 
1  tailor  makes   ^         suits  in  19  days. 

494 

To  do  494  suits  in  19  days  will  take  JQ — TQ  tailors. 

-Lo  x  j.  y 

12x7 
2 

n 

494 


13x19 


Ans. 


12  x  7 

46.  If  a  family  of  9  persons  spend  $306  in  4 
months,  how  many  dollars  would  a  faro  ly  of  15 
persons  spend,  at  the  same  rate,  in  8  months  ? 

Ans.  11020. 

47.  If  20  bushels  of  wheat  are  sufficient  for  a 
family  of  15  persons  3  months,  how  much  will  be 
sufficient  for  4  persons  9  months  ?     Ans.  16. 

48.  If  7  men  can  build  36  rods  of  wall  in  3  days, 
how  many  rods  can  20  men  build  in  14  days  ? 

49.  If  7  men  can  reap  84  acres  of  wheat  in  12 
days,  how  many  men  can  reap  100  acres  in  5  days  ? 


248  Cancellation  and  Analysis*         [§  10. 

50.  If  18  men  can  build  a  wall  40  rods  long, 
5  ft.  high,  and  4  ft.  thick  in  15  days,  in  what  time 
will  20  men  build  a  wall  87  rods  long,  8  ft.  high, 
and  5  ft.  thick  ? 

51.  How  many  yards  of  flannel  that  is  1^  yards 
wide  will  line  a  cloak  containing  9  yards  of  cloth 
that  is  |  of  a  yard  wide.     Ans.  4|  yds. 

52.  A  regiment  of  soldiers,  consisting  of  1000 
men,  is  to  be  supplied  with  new  coats  ;  each  coat  is 
to  contain  2|  yards  of  cloth  1J  yards  wide,  and  is 
to  be  lined  with  flannel  f  of  a  yard  wide.     How 
many  yards  of  flannel  will  be  required  ? 

53.  A  ship's  crew  of  18  men  has  enough  pro- 
visions to  last  the  voyage,  if  each  man  is  allowed . 
20  oz.  per  day.     If  a  shipwrecked  crew  of  6  per- 
sons is  picked  up,  what  must  then  be  the  daily 
allowance  of  each  person  ? 

Suggestion :  18  x  20  oz.  per  day  is  to  be  divided 
among  24  men. 

54.  If  8  boarders  will  eat  a  quantity  of  flour 
in  15  days,  how  long  will  it  last  if  4  more  boarders 
join  them  ?     [How  long  would  it  last  one  man  ? 
how  long  would  it  last  12  men  ?] 

55.  Suppose  650  men  are  in  a  garrison,  and 
have  provisions  enough  to  last  them  two  months ; 
how  many  men  must  leave  the  garrison  in  order  to 
have  the  provisions  last  those  who   remain   five 
months?     Ans.  390. 

56.  If  a  staff  4  feet  long  cast  a  shadow  on  level 
ground  6  ft.  8  in.  long,  what  is  the  height  of  a 
steeple  which  casts  a  shadow  175  feet  long  at  the 
same  time?     Ans.  105  feet. 


Cancellation  and  Analysis.  249 

57.  If  a  man  travels  64  rods  in  .05  of  an  hour, 
how  many  minutes  will  it  take  him  to  travel  a 
mile? 

58.  A  man  receives  $  18  for  six  days'  work  of  8 
hours  each;   what  should  he  receive  for  5  days' 
work  of  9  hours  each  ? 

59.  If  6  men  can  build  20  feet  of  a  stone  wall 
in  10  days,  how  many  men  can  build  360  feet  of 
the  same  wall  in  90  days  ? 

60.  If  3  men  can  build  a  wall  60  feet  long,  8 
feet  high,  and  3  feet  thick  in  64  days  of  9  hours 
each,  how  many  days  of  8  hours  each  will  20  men 
require  to  build  a  wall  400  feet  long,  9  feet  high, 
and  5  feet  thick  ? 

61.  If  6  men  can  build  a  wall  80  feet  long,  10 
feet  high,  and  9  feet  thick,  in  100  days  of  9  hours 
each,  how  many  days  of  10  hours  each  will  be  re- 
quired by  15  men  to  build  a  wall  200  feet  long,  9 
feet  high,  and  5  feet  thick  ? 

62.  If  4  men  dig  a  trench  84  feet  long  and  5 
feet  wide  in  3  days  of  8  hours  each,  how  many 
men   can  dig  a   trench  of   the   same   depth  420 
feet  long  and  3  feet  wide  in  4  days  of  9  hours 
each  ? 

63.  If  496  men,  in  5  days  of  12  h.  6m.  each,  dig 
a  trench  of  9  degrees  of  hardness,  465  feet  long, 
3 1  feet  wide,  and  4|  feet  deep,  how  many  men  will 
be  required  to  dig  a  trench  2  degrees  of  hardness, 
168|  feet  long,  7^  feet  wide,  and  2|  feet  deep  in 
22  days  of  9  hours  each  ?     Aiis.  15  men. 

64.  A  man  has  a  bin  7  ft.  long,  2^  ft.  wide, 


250  Cancellation  and  Analysis.         [§  10. 

and  2  ft.  deep,  which  contains  28  bushels  of  corn. 
How  deep  must  he  build  another  bin,  which  is  to 
be  18  ft.  long,  1  ft.  10|  in.  wide,  in  order  to  con- 
tain 120  bushels?  Ans.  4  ft.  5£  in. 

65.  Two  men,  A  and  B,  traded  in  company  ; 
A  furnished  f  of  the  stock  and  B  £  ;   they  gained 
$864.00 ;    what    was    each    man's   share    of    the 
gain  ? 

66.  Three  men,  A,  B,  and  C,  traded  in  com- 
pany; A  furnished  |J  of  the  capital,  B  J|,  and  C 
the  rest.     They  gained  18,453.28  ;  what  was  each 
one's  share  of  the  gain  ? 

67.  Two  men,  B  and  C,  bought  a  barrel  of  flour 
together.     B  paid  $5.00  and  C  13.00 ;  what  part 
of  the  whole  price  did  each  pay  ?     What  part  of 
the  flour  ought  each  to  have  ? 

68.  Three  men,  C,  D,  and  E,  traded  in  com- 
pany;  C   put  in  $855,  D  $945,  and  E  $1179; 
how  many  dollars  did  they  all  put  in  ?    What  part 
of   the    whole   did    each   put   in?      They   gained 
$1340.55;    what   was   each   man's   share   of   the 
gain? 

69.  Five  men,  A,  B,  C,  D,  and  E,  freighted  a 
vessel:  A  put  on  board  goods  to  the  amount  of 
$4000,  B  $15,000,  C  $11,000,  D  $7500,  and  E 
$850.     During  a  storm  the  captain  was  obliged 
to    throw    overboard    goods    to    the    amount    of 
$13,039 ;  what  was  each  man's  share  of  the  loss  ? 

70.  Three  men  hired  a  pasture  for  $42.00 ;  the 
first  put  in  4  horses  ;  the  second  6  ;  and  the  third 
8.     What  ought  each  to  pay  ? 


Cancellation  and  Analysis.  251 

71.  A  man  failing  in  trade  owes  A  12700,  B 
$1800,  C  $1500  ;  and  he  has  only  $2100  in  prop- 
erty, which  he  agrees  to  divide  among  his  creditors 
in  proportion  to  the  several  debts.    What  will  each 
receive  ? 

72.  During  a  storm  a  master  of  a  vessel  was 
obliged  to  throw  overboard  -^  of  the  whole  cargo. 
What  part  of  the  whole  cargo   did   a  man    lose 
who  owned  f  of  it  ? 

o 

73.  A  man  owned  -f^  of  the  capital  of  a  cotton 
manufactory,  and  sold   T\    of   his   share.      What 
part  of  the  whole  capital  did  he  sell  ?     What  part 
did  he  then  own  ? 

74.  How  many  bushels  of  apples,  at  \  of  a  dol- 
lar per  bushel,  may  be  bought  for  f  of  a  dollar  ? 
How  many  at  f  of  a  dollar  per  bushel  ? 

75.  Two   men  bought  a  barrel  of   flour :    one 
gave  2 1  dollars  and  the  other  3f  dollars;    what 
did  they  give  for  the  whole  barrel?     What  part 
of  the  whole  value  did  each  pay  ?     What  part  of 
the  flour  should  each  have  ? 

76.  Two  men  hired  a  pasture  for  21  dollars. 
One  kept  his  horse  in  it  5^  weeks,  and  the  other 
7  :t  weeks  ;  what  ought  each  to  pay  ? 

77.  A  man  being  asked  his  age  answered,  that 
he  was  24  years  old  when  he  was  married,  and  that 
he  had  lived  with  his  wife  f  of  his  whole  life. 
What  was  his  age  ? 

78.  A  person  having  |  of  a  vessel  sold  f  of  his 
share  for  $8,400.00  ;  what  part  of  the  whole  vessel 
did  he  sell  ?     What  was  the  whole  vessel  worth  ? 


252  Cancellation  and  Analysis.         [§  10. 

79.  If  |  of  a  ship  is  worth  ^  of  her  cargo,  and 
the  cargo  is  valued  at  2100c£,  what  is  the  value  of 
the  ship  ? 

80.  There  is  a  pole  standing  so  tnat  |-  of  it  is  in 
the  water,  |  as  much  in  the  mud  as  in  the  water, 
and  7 §  feet  above  the  water.     What  is  the  whole 
length  of  the  pole  ? 

81.  Two  men,  A  and  B,  having  found  a  bag  of 
money,  disputed  who  should  have  it.     A  said  .] ,  .1 , 
and  I  of  the  money  made  130  dollars,  and  if  B 
could  tell  him  how  much  there  was  in  the  bag  he 
should  have  all  the  money,  otherwise,  he  should 
have  nothing.     How  much  was  there  in  the  bag  ? 

82.  A  merchant  sold   a  quantity  of  goods  for 
$3,846,  by  which  bargain  he  gained  £  of  the  first 
cost.     What  was  the  first  cost,  ancWiow-rnuch  did 
he  gain  ? 

83.  A  merchant  sold  a  bale  of  cloth  for  1351, 
by  which  he  gained  -^  of  what  it  cost  him.     How 
much  did  it  cost  him,  and  how  much  did  he  gain  ? 

84.  A  merchant  sold  a  quantity  of  flour  for 
1143.00,  by  which  he  gained  |  of  the  cost.     How 
much  did  it  cost,  and  how  much  did  he  gain  ? 

85.  A  merchant  sold  a  quantity  of  goods  for 
1187.00,  by  which  he  lost  |  of  the  first  cost.     How 
much  did  it  cost,  and  how  much  did  he  lose  ? 

86.  A  merchant  sold  a  quantity  of  molasses  for 
$259.00,  by  which  he  lost   |   of  the  cost.     How 
much  did  it  cost,  and  how  much  did  he  lose  ? 

87.  A  merchant  sold  a  quantity  of  goods  for 
$946,  by  which  he  lost  T\  of  the  cost.     How  much 
did  he  lose  ? 


Cancellation  and  Analysis.  253 

88.  A  farmer  mixed  15  bushels  of  rye,  at  64 
cents  per  bushel ;  18  bushels  of  corn  at  55  cents 
per  bushel ;  and  21  bushels  of  oats,  at  28  cents 
per  bushel.     How  many  bushels  were  there  of  the 
mixture  ?      What  was  the  whole  worth  ?      What 
was  it  worth  per  bushel  ? 

89.  A  grocer  mixed  123  Ibs.  of  sugar  that  was 
worth  8  cents  per  pound ;  87  Ibs.  that  was  worth 
11  cents  per  pound  ;  and  15  Ibs.  that  was  worth  13 
cents  per  pound.     What  was  the  mixture  worth 
per  pound  ? 

90.  Three  merchants,  A,  B,  and  C,  freight  a 
ship  with  coal.     A  puts  on  board  500  tons,  B  340, 
and  C  94  ;   in  a  storm  they  are  obliged  to  cast 
150  tons  overboard.     What  loss  does  each  sus- 
tain? 

91.  Two  men  hired  a  pasture  for  136.     A  put 
in  3  horses  for  4  months,  and  B  5  horses  for  3 
months.     What  ought  each  to  pay? 

Suggestion :  3  horses  for  4  months  is  the  same 
as  4  times  3  or  12  horses  for  1  month ;  and  5 
horses  for  3  months  is  the  same  as  3  times  5  or  15 
horses  for  1  month. 

92.  Four  men  jointly  hired  a  pasture  for  f  1.00. 
A  turned  in  7  oxen  for  12  days,  B  9  oxen  for  14 
days,  C  11  oxen  for  25  days,  and  D  15  oxen  for 
37  days.     How  much  ought  each  to  pay  ? 

93.  Three  men  entered    into  partnership,  and 
traded  as  follows :  A  put  in  $150,  and  at  the  end 
of  7  months  took  out  f  50  ;  5  months  after  he  put 
in  $170  ;  B  put  in  $205,  and  at  the  end  of  5  months 


254  Cancellation  and  Analysis.         [§10. 

he  put  in  $110  more,  but  4  months  after  took  out 
$150  ;  C  put  in  $300,  and  when  8  months  had 
elapsed  he  drew  out  $150.00  ;  but  5  months  after 
he  put  in  $500.00.  Their  partnership  continued  18 
months,  at  the  end  of  which  time  they  had  gained 
$660.  Required  each  person's  share  of  the  gain. 

94.  A  owes  B  a  sum  of  money,  of  which  J  is  to 
be  paid  in  2  months,  J  in  3  months,  and  the  rest 
in  6  months.     If  A  prefers  to  pay  the  whole  sum 
at  the  same  time,  when  should  he  pay  it?      [A 
method  of  solution  may  perhaps  be  suggested  by 
first  solving  the  problem  on  the  supposition  that 
the  entire  sum  is  $6.] 

95.  Two  men  were  talking  of  their  ages ;  one 
said,  "  |  of  my  age  is  equal  to  \  of  yours ;  and  the 
sum  of  our  ages  is  95."     What  were  their  ages  ? 

Suggestion :  -^5  of  the  second  equals  \  g  of  the 
first ;  therefore  the  second  =  -^  of  first,  therefore 
first  +  second  =  -1-jj-  of  first  =  95. 

96.  If  a  man  can  do  f  of  a  piece  of  work  in  one 
day,  in  what  part  of  a  day  can  he  do  |  of  it? 
How  long  will  it  take  him  to  do  the  whole  ? 

97.  A  farmer  hired  two  men  to  mow  a  field ; 
one  of  them  could  mow  ^  of  it  in  a  day,  and  the 
other  I  of  it.     What  part  of  it  would  they  both 
together  do  in  a  day  ?     How  long  would  it  take 
them  both  to  mow  it  ? 

98.  A  gentleman  hired  3  men  to  build  a  wall ; 
the  first  could  build  it  alone  in  8  days,  the  second 
in  10  days,  and  the  third  in  12  days.     What  part 
of  it  could  each  build  in  a  day  ?     How  long  would 
it  take  them  all  together  to  build  it  ? 


Cancellation  and  Analysis.  255 

99.  A  can  do  a  certain  piece  of  work  in  10 
days,  working  8  hours  a  day.  B  can  do  the  same 
work  in  9  days,  working  12  hours  a  day.  They 
decide  to  work  together,  and  to  finish  the  work  in 
6  days.  How  many  hours  a  day  must  they  work  ? 

100.  A  man  and  his  wife  found  that  when  they 
were  together  a  bushel  of  corn  would  last  15  days, 
but  when  the  man  was  absent,  it  would  last  the 
woman  alone  27  days.     What  part  of  it  did  both 
together  consume  in  a  day  ?     What  part  did  the 
woman  alone  consume  ?     What  part  did  the  man 
alone  consume  ?     How  long  would  it  last  the  man 
alone  ? 

101.  A  cistern  has  3  cocks  to  fill  it,  and  one  to 
empty  it.     The  first  cock  will  fill  it  alone  in  3 
hours,  the  second  in  5  hours,  and  the  third  in  9 
hours.      The  other  will  empty  it  in  7  hours.     If 
all  the  cocks  are  allowed  to  run  together,  in  what 
time  will  it  be  filled  ? 

102.  Divide  25  apples  between  two  persons  so 
as  to  give  one  seven  more  than  the  other. 

Suggestion :  Give  one  of  them  7,  and  then  di- 
vide the  rest  equally. 

103.  A  gentleman  divided  an  estate  of  115,000 
between  his  two  sons,  giving  the  elder  $2500  more 
than  the  younger.     What  was  the  share  of  each  ? 

104.  A    gentleman    bequeathed    an    estate    of 
$ 40,000  to  his  wife,  son,  and  daughter:  to  his  wife 
he  gave  $1500  more  than  to  the  son,  and  to  the 
son  13500  more  than  to  the  daughter.     What  was 
the  share  of  each  ? 


256  Cancellation  and  Analyst*.        [§  10. 

105.  A,  B,  and  C    built    a   house    which   cost 
$35,000  ;  A  paid  1500  more  than  B,  and  C  $300 
less  than  B.     What  did  each  pay  ? 

106.  A  man  bought  a  sheep,  a  cow,  and  an  ox 
for  $82  ;  for  the  cow  he  gave  $10  more  than  for 
the  sheep  ;  and  for  the  ox  $10  more  than  for  both. 
What  did  he  give  for  each  ? 

107.  A  man  sold  some  calves  and  some  sheep 
for  $216 ;  the  calves  at  $10,  and  the  sheep  at  $16 
apiece.      There   were   twice   as    many   calves    as 
sheep.     What  was  the  number  of  each  sort  ? 

Suggestion:   There  were  two  calves   and   one 
sheep  for  every  $36. 

108.  A  farmer   drove   to    market    some    oxen, 
some  cows,   and  some  sheep,   which  he   sold   for 
$1498 ;  the  oxen  at  $56,  the  cows  at  $34,  and  the 
sheep  at  $15.     There  were  twice  as  many  cows  as 
oxen,   and   three   times  as  many  sheep   as  cows. 
How  many  were  there  of  each  kind  ? 

109.  Said  A  to  B,  my  horse  and  saddle  together 
are  worth  $150  ;  but  my  horse  is  worth  9  times  as 
much  as  the  saddle.    What  was  the  value  of  each  ? 

110.  A  man  driving  some  sheep  and  some  cat- 
tle, being  asked  how  many  he  had  of  each  kind, 
said   he    had   174   in   all,   and  there  were   ^  as 
many  cattle  as  sheep.     Required  the  number  of 
each  kind. 

111.  A  gentleman  left  an  estate  of  $13,000  to 
his  four  sons,  in  such  a  manner  that  the  third  was 
to  have  1|  as  much  as  the  fourth,  the  second  as 
much  as  the   third  and  fourth,  and   the  first  as 


Cancellation  and  Analysis.  257 

much  as  the  other  three.     What  was  the  share  of 
each  ? 

112.  Three   persons,   A,  B,  and  C,  traded   in 
company.     A  put  in  $75,  B  $ 40.      They  gained 
$64,  of  which  C  took  $18  for  his  share.     What 
did  C  put  in  ? 

113.  A  person  buys  12  apples  and  6  pears  for 
17  cents,  and  afterwards  3  apples  and  12  pears  for 
20  ceiits.     What  is  the  price  of  an  apple,  and  of  a 
pear  ? 

Suggestion :  At  the  second  time  he  bought  3 
apples  and  12  pears  for  20  cents.  4  times  all  this 
will  make  12  apples  and  48  pears  for  80  cents  : 
the  price  of  12  apples  and  6  pears  being  taken 
from  this,  will  leave  63  cents  'for  42  pears,  which 
is  1J  cent  apiece. 

114.  Two  persons  were  talking  of  their  ages ; 
one  said,  "  |  of  mine  is  equal  to  |  of  yours,  and  the 
difference  between  our  ages  is  10  years."     What 
were  their  ages  ? 

115.  A  man  having  $100  spent  a  part  of  it ;  he 
afterwards    received   five   times   as   much    as    he 
spent,  and  then  his  money  was  double  what  it  was 
at  first.     How  much  did  he  spend  ? 

116.  A,  B,  and  C  hire  a  pasture  for  $92.     A 
pastures  6  horses  for  8  weeks,  B  12  oxen  for  10 
weeks,  and  C  50  cows  for  12  weeks.     Now,  if  5 
cows  are  reckoned   as  3  oxen,  and   3  oxen  as  2 
horses,  how  much  should  each  man  pay  ? 

117.  By  a  pipe  of  a  certain  capacity  a  cistern 
can  be  emptied  in  3y^  hours  ;  in  what  time  can  it 


258  Cancellation  and  Analyxix. 

be  emptied  by  a  pipe  the  capacity  of  which  is  | 
greater  ? 

118.  A  and  B,  44  miles  apart,  travel  towards 
each  other.     A  travels  -^  of  the  whole  distance, 
v;hile  B  travels  |  of  the  remainder.     How  far  are 
they  then  apart  ? 

119.  A  can  do  a  piece  of  work  in  10  days,  A 
and  C  can  do  it  in  7  days,  A  and  B  can  do  it  in  6 
days  :  in  how  many  days  can  B  and  C  together  do 
it? 


APPENDIX. 


CHAPTER  I. 
Roman  Notation. 

The  figures  that  are  ordinarily  used  to  represent 
numbers  are  called  Arabic  figures,  because  they  were 
first  introduced  into  Europe  by  the  Arabs,  who  had 
derived  them,  however,  from  Hindostan.  Numbers  are 
also  sometimes  represented  by  Roman  letters,  as  indi- 
cated in  the  following  table  : 

One, 
Two, 

Three, 

Four, 

Five, 

Six, 

Seven, 

Eight, 

Nine, 

Ten, 

Eleven, 

Twelve, 

Thirteen, 

Fourteen, 

Fifteen, 


I. 

Sixteen, 

XVI. 

II. 

Seventeen, 

XVII. 

III. 

Eighteen, 

XVIII. 

IV. 

Nineteen, 

XIX. 

V. 

Twenty, 

XX. 

VI. 

Twenty-one, 

XXI. 

VII. 

Twenty-two, 

XXII. 

VIII. 

Twenty-three, 

XXIII. 

IX. 

Twenty-four, 

XXIV. 

X. 

Twenty-five, 

XXV. 

XL 

Twenty-six, 

XXVI. 

XII. 

Twenty-seven, 

XXVII. 

XIII. 

Twenty-eight, 

XXVIII. 

XIV. 

Twenty  -nine, 

XXIX. 

XV. 

Thirty, 

XXX. 

260 


Appendix. 


[Ch.  1. 


Thirty-one, 

XXXI. 

Two  hundred, 

CC. 

Thirty-two, 

XXXII. 

Three  hundred, 

ccc. 

Forty, 

XL. 

Four  hundred, 

cccc. 

Fifty, 

L. 

Five  hundred, 

D. 

Sixty, 

LX. 

Six  hundred, 

DC. 

Seventy, 

LXX. 

Seven  hundred, 

DCC. 

Eighty, 

LXXX. 

Eight  hundred, 

DCCC. 

Ninety, 

XC. 

Nine  hundred, 

DCCCC. 

One  hundred, 

C. 

One  thousand, 

M. 

One  thousand  eight  hundred  and  twenty-six, 
MDCCCXXVI. 

Note  1.  The  following  description  of  the  development  of  the 
Roman  Notation,  taken  from  Warren  Colburn's  Arithmetic,  A 
Sequel  to  Intellectual  Arithmetic  (see  page  109),  may  be  found  in- 
teresting, and  will  tend  to  fix  the  notation  in  mind. 

One  was  written  with  a  single  mark,  thus :  / 
Two  was  written  with  two  marks,  // 

Three  was  written  /// 

Four  was  written  //// 

Five  was  written  ///// 

Six  was  written  II I  III 

Seven  was  written  II Hill 

Eight  was  written  II I  Hill 

Nine  was  written  ///////// 

Ten,  instead  of  being  written  with  ten 
marks,  was  expressed  by  two  marks 
crossing  each  other,  thus,  X 

Two  tens,  or  twenty,  were  written  XX 

Three  tens,  or  thirty,  were  written  XXX 

And  so  on  to  ten  tens,  which  were  written  with  ten  crosses.  But 
as  it  was  found  inconvenient  to  express  numbers  so  large  as  seven 
or  eight  with  marks  as  represented  in  the  foregoing,  the  X  was 
cut  in  two,  thus,  X,  and  the  upper  part  V  was  used  to  express 
one  half  of  ten,  or  five,  and  the  numbers  from  five  to  ten  were  ex- 
pressed by  writing  marks  after  the  V,  to  express  the  number  of 
units  added  to  five. 


Roman  Notation.  261 

Six  was  written  ]/f 

Seven  was  written  ]/H 

Eight  was  written  J7// 

Nine  was  written  /)( 

Eleven  was  written  XI 

Twelve  was  written  XII 

Twenty-seven  was  written    XXVII 

To  express  ten  X's,  or  ten  tens,  or  one  hundred,  three  marks 
were  used,  thus,  C ;  and  to  avoid  the  inconvenience  of  writing 
seven  or  eight  X's,  the  C  was  divided,  thus,  H,  and  the  lower  part, 
L,  used  to  express  five  X's,  or  fifty. 

To  express  ten  hundreds,  four  dashes  were  used,  thus,  I VI.  This 
last  was  afterwards  written  in  this  form  CD,  and  sometimes  C  ID, 
and  was  then  divided,  and  ID  was  used  to  express  five  hundreds. 
These  dashes  resemble  some  of  the  letters  of  the  alphabet, 
which  letters  were  afterwards  substituted  for  them.  I  resembles 
I ;  V  resembles  V  ;  X  resembles  X  ;  L  resembles  L  ;  C  was  sub- 
stituted for  C;  ID  resembles  D;  and  I VI  resembles  M.  The 
numbers  four,  nine,  forty,  and  ninety  were  afterwards  denoted 
more  briefly,  as  shown  below. 

IIII  by  IV  XXXX  by  XL 

VIIII  by  IX  LXXXX  by  XC 

Writing  an  I  before  a  V  or  an  X  decreases  the  value  of  the  V 
or  X  by  I ;  and  writing  an  X  before  an  L  or  a  C  decreases  the 
value  of  the  L  or  C  by  X.  Wherever  these  numbers  (four,  nine, 
forty,  and  ninety)  occurred  they  were  abbreviated  in  the  same  way. 
Thus  XIIII  became  XIV,  XVIIII  became  XIX,  LVIIII  became 
LIX,  CLXXXX  became  CXC,  etc. 

Note  2.  The  explanation,  substantially  as  given  in  Note  1,  of 
the  origin  of  the  characters  of  the  Roman  Notation,  was  advo- 
cated by  Sir  John  Leslie  in  his  Philosophy  of  Arithmetic  (1820), 
and  by  some  writers  of  the  16th  and  17th  centuries.  On  this 
point,  however,  the  Encyclopaedia  Britannica  (Ninth  Edition), 
says  :  "  This  explanation  is  perhaps  too  ingenious.  .  .  •  One  does 
not  readily  see  how  the  C  could  be  formed  from  the  X  or  the  M 
from  the  C  ;  and  it  appears  far  more  likely  that  the  signs  for 
100  and  1000  are  merely  the  initial  letters  of  Centum  and  Mille. 
...  In  any  case  the  V,  L,  and  D  appear  to  be  respectively  the 
halves  of  X,  the  angular  C  (C)  and  the  rounded  M  (po)  [fre- 
quently written  C  ID]." 


262  Appendix.  [Ch.  2. 

EXAMPLES. 

1.  Read    the    numbers  XLIV,    XXXIX,    CXIX, 
CLIII,     LXXXVI,     LX,     XXI,     Mil,     DXCIX, 
MDCCCLXXXVIL 

2.  Denote  by  Roman  letters  the  numbers  thirty-eight, 
eighty-four,  thirteen,  four  hundred  and  thirty-two,  one 
thousand    and    seventy -two,     eighteen    hundred    and 
seventy-five. 


APPENDIX.     CHAPTER  II. 
THE  METRIC*  SYSTEM  OF  MEASURES. 

In  1799  the  French  Legislature  adopted  a  fixed 
length,  which  it  agreed  to  call  a  metre  (a  word  derived 
from  the  Greek  metron,  a  measure),  as  the  standard 
unit  of  linear  measure.  According  to  the  best  measure- 
ments that  could  then  he  obtained,  this  fixed  length 
(called  in  English  a  meter  f)  was  the  ten-millionth  part 
of  the  distance  from  the  equator  to  the  north  pole ;  it 
was  chosen  because  it  appeared  to  meet  the  demand  for 
an  invariable  unit  of  length  which  could  be  re-determined 
from  the  definition  at  any  future  time. 

There  was  then  constructed,  with  the  meter  as  a 
basis,  the  so-called  Metric  System  of  Measures,  which 
will  be  described  in  the  following  pages. 

The  metric  system  of  measures  has  also  been  adopted 
in  Germany,  Spain,  Portugal,  Belgium,  Holland,  Switzer- 
land, Italy,  Austria,  Sweden,  Denmark,  Greece,  British 
India,  Turkey,  Mexico,  Brazil,  and  other  States  of  South 
America,  and  is  in  use  to  some  extent  in  this  and  most 
other  countries. 

*  Pronounced  me'tric.  t  Pronounced  mee'-ter. 


A.]  The  Metric  System.  263 

A.  Linear  Measure. 

In  the  metric  system  of  measures  the  principal  meas- 
ure or  unit  of  length  is  the  meter,  a  measure  very  nearly 
39.37  inches  long.*  The  smaller  measures  of  length 
are  one  tenth,  one  hundredth,  and  one  thousandth  of  a 
meter;  and  the  larger  measures  of  length  are  ten,  one 
hundred,  one  thousand,  and  ten  thousand  meters. 

The  names  of  these  measures  are : 

MILLIMETER  f  (mm)  for  T73^ ^  of  a  meter. 
The  prefix  milli  denotes  thousandth. 

CENTIMETER  |  (cm)  for  T J^  of  a  meter. 
The  prefix  centi  denotes  hundredth. 

DECIMETER  f  (dm)  for  T^  of  a  meter. 
The  prefix  deci  denotes  tenth. 

METER  (m)  for  the  principal  unit. 

DEKAMETER  (Dm)  for  10  meters. 
The  prefix  deka  denotes  ten. 

HECTOMETER  (Hm)  for  100  meters. 

The  prefix  hecto  denotes  hundred. 

KILOMETER  (Km)  for  1000  meters. 
The  prefix  kilo  denotes  thousand. 

MYRIAMETER  (Mm)  for  10000  meters. 

The  prefix  myria  denotes  ten  thousand. 
1.  Complete   the    following   table   by  filling   in   the 

*  The  standard  meter  is  a  platinum  bar  carefully  preserved  at 
Paris ;  and  every  meter-stick  anywhere  in  use  should  be  of  the 
same  length  as  the  standard.  The  length  of  the  standard  meter, 
expressed  in  inches,  is  39.3707904  ;  it  is  sufficiently  accurate  for 
most  practical  purposes  to  say  that  a  meter  contains  39.37  inches. 

t  Pronounced  millimeter,  centimeter,  decimeter.  All  the  metric 
names  are  accented  on  the  first  syllable. 


264  Appendix.  [Ch.  2. 

proper  numbers  on  the  right,  and  the  proper  abbreviations 
within  the  parentheses  : 

1  millimeter  (mm)  =  .001  meter 

1  centimeter  (       )  = 

1  decimeter  (       )  = 

1  meter  (  m  )  =  1  meter 

1  dekameter  (       )  = 

1  hectometer  (       )  =  r-1° 

1  kilometer  (       )  = 

1  myriameter  (       )  = 

2.  Complete  the  following  table  by  filling  in 
the  proper  numbers  on  the  left  and  the  proper 
abbreviations  within  the  parentheses : 

10  millimeters  =1  centimeter  (cm) 

centimeters  =  1  decimeter  (  ) 

decimeters    =1  meter  (  ) 

meters          =  1  dekameter  (  ) 

dekameters  =  1  hectometer  (  ) 

hectometers  =  1  kilometer  (  ) 

kilometers    =1  myriameter  (  ) 

3.  Make  a  decimeter  rule  or  measure,  by 
drawing  with  a  fine  pen  on  the  edge  of  a  piece 
of  cardboard  the  figure  given  on  the  right. 

4.  By  the  aid  of  the  measure  just  made 
find  (a)  the  nearest  number  of  decimeters  in 
the  length  of  the  cover  of  this  book  (Z>),  the 
nearest  number  of  centimeters. 

Find  (a)  the  nearest  number  of  centimeters 
in  the  thickness  of  this  book  (&),  the  nearest 
number  of  millimeters. 

5.  Find  the  nearest  number  of  centimeters 
in  the  length  of  your  lead  pencil. 


.4 


A.]  The,  Metric  System.  265 

6.  Find  (a)  the  nearest  number  of  decimeters  in  the 
width  of  the  nearest  window ;  (b)  the  nearest  number  of 
centimeters. 

7.  What  is  the  nearest  number  of  decimeters  in  the 
distance  of  the  window-sill  from  the  floor  ? 

8.  Measure   the   length   and  width   of   a  window- 
pane  and  find  the  nearest  number  of  centimeters  in  each. 

9.  Find  the  nearest  number  of  decimeters  in  the 
width  of  your  desk. 

10.  Find  the  nearest  number  of  decimeters  in  the 
distance  of  the  highest  part  of  your  chair  from  the  floor. 

1 1.  What  is  the  nearest  number  of  centimeters  in  the 
distance  from  the  point  of  your  elbow  to  the  tip  of  your 
middle  finger  ? 

1 2.  With  a  stick  or  a  string  make,  by  the  aid  of  your 
decimeter  rule,  a  meter  rule. 

1 3.  a.  How  long  is  the  room  that  you  are  now  in  ? 
[Find  first  how  many  whole  meters  there  are  and  then 
the  nearest  number  of  centimeters  that  are  left  over.] 

b.  How  many  centimeters  are  there  in  the  length  of 
the  room  ?  c.  How  many  steps  do  you  take  in  walking 
from  one  end  of  the  room  to  the  other  ?  d.  What  is 
the  nearest  number  of  centimeters  in  the  length  of  each 
step  if  all  the  steps  are  equally  long  ? 

14.  a.  How   many    steps    do   you   take  in  walking 
across  the  room  ?     b.  How  wide  then  is  the  room  if  the 
length  of  each  step  is  the  same  as  before  ?     c.  Measure 
the  width  of  the  room  with  your  rule  and  see  how  much 
the  result  differs  from  that  just  found. 

15.  Is  the  length  of  the  room  more  or  less  than  a 
dekameter  ?     how  much  ?     [Give  the  answer  in  centi- 
meters.] 

16.  Is  the  width  of  the  room  more  or  less  than  a 
dekameter  ?  how  much  ? 


266  Appendix.  [Ch.  2. 

17.  a.  How  many  steps  do  you  take  in  walking  the 
length  of  the  building  that  you  are  now  in  ?  how  many 
meters  long  is  the  building  if  the  length  of  each  step  is 
the  same  as  found  in  Example  13  d  ? 

b.  Find  in  the  same  way  the  approximate  width  of 
the  building. 

18.  How  many  millimeters  are  there  in  a  centimeter  ? 
how   many   centimeters    in    a   decimeter  ?    how   many 
decimeters  in  a  meter  ?   how  many  meters  in  a  deka- 
meter  ? 

19.  How  many  decimeters  are  there  in  3  Dm,  2  m, 
and  6  dm?  (Ans.  326)  how  many  meters?  (Ans.  32.6) 
how  many  dekameters  ?  (Ans.  3.26) 

20.  How  many  centimeters  are  there  in  4  m,  3  dm, 
and  8  cm  ?  how  many  decimeters  ?  how  many  meters  ? 
how  many  dekameters  ? 

21.  How  many  hectometers  are  there  in  1.6046  Km  ? 
how  many  dekameters  ?  how  many  meters  ?  how  many 
decimeters  ? 

22.  How  many  decimeters  are  there  in  1632  cm? 
how  many  meters  ?  how  many  dekameters  ? 

23.  How  many  centimeters  are  there  in  736.8  mm  ? 
how  many  decimeters  ?  how  many  meters  ? 

24.  If  a  building  is  123  m  long,  what  measure  is  used 
when  its  length  is  denoted  by  12.3  ?  by  1.23  ?  by  0.123  ? 
by  1230  ?  by  12300  ? 

25.  a.  How  many  meters  are  there  in  6  Km  ?  in  3 
Hm  ?  in  8  Dm  ? 

b.  How  many  meters  are  there  in  8  mm  ?  in  9  cm  ? 
in  4  dm  ? 

26.  How  many  millimeters  are  there  in  2  dm,  3  cm, 
and  6  mm  ?  How  many  meters  ? 

27.  How  many  meters  are  there  in  6  Hm  and  2 


B.]  The  Metric  System.  267 

28.  If  there  are  39.37  inches  in  a  meter  how  many 
inches  are  there  in  a  decimeter  ?  how  many  in  a  centi- 
meter ? 

29.  How  many  yards  are  there  in  a  meter  ? 

Ans.  1.09  4- yds. 

30.  How  many  rods  are  there  in  a  dekameter  ? 

31.  How  many  miles  are  there  in  a  kilometer  ? 

Ans.  .62  miles. 

32.  How  many  centimeters  are  there  in  an  inch  ? 

33.  How  many  decimeters  are  there  in  a  foot  ? 

34.  How  many  meters  are  there  in  a  yard  ? 

35.  How  many  dekameters  are  there  in  a  rod  ? 

36.  How  many  kilometers  are  there  in  a  mile  ? 

Ans.  1.60  +  Km. 

37.  How  long  a  strip  of  paper  as  wide  as  this  page 
would  be  required  for  200  pages  of  this  book  ?     [Find 
the  answer  first  in  centimeters  and  then  change  it  to 
meters.] 

38.  At  $1.75  a  meter  what  will  a  kilometer  of  silk 
cost? 

39.  How  far  must  I  walk  in  going  from  A  to  B,  a 
distance  of  6  Hm  3  m,  and  then  from  B  to  C,  a  dis- 
tance of  9  Dm  8  m  ?    Ans.  1  Hm  1  m. 

B.  Square  Measure. 

1.  The  figure  in  the  margin   is  a  square 
centimeter  (sq.  cm).     How  many  square  mil- 
limeters does  it  contain  ? 

2.  How  many  square  centimeters  are  there 

in  a  figure  4  centimeters  long  and  2  centimeters  wide  ? 
how  many  in  a  figure  twice  as  long  and  twice  as  wide  ? 

3.  How  many    square   centimeters   are   there   in   a 
figure  4  centimeters  long  and  4  centimeters  wide  ?  how 


268  Appendix.  [Ch.  2. 

many  in  a  figure  8  centimeters  long  and  2  centimeters 
wide  ?  How  much  more  string  is  required  to  surround 
the  second  figure  than  the  first  ? 

4.  Draw  on  paper  as  accurately  as  you  can  a  square 
decimeter    (sq.  dm).      How  many   square   centimeters 
does  it  contain  ? 

5.  How  many  square  decimeters  are  there  in  a  square 
meter  (sq.  m)  ? 

6.  How  many  square  meters  are  there  in  a  square 
dekameter  (sq.  Dm)  ? 

[The  square  dekameter,  called  also  an  ar*  (a),  is  a 
convenient  measure  to  use  in  measuring  land.  The 
square  meter  is  also  called  a  centar  (ra).] 

7.  How  many  square  dekameters  or  ars  are  there  in 
a  square  hectometer  ? 

[The  square  hectometer  is  also  called  a  hectar  (Aa).] 

8.  How   many   square    hectometers   or   hectars    are 
there  in  a  square  kilometer  (sq.  km)  ? 

9.  Complete  the  following   table   hy   filling   in   the 
proper  numbers  on  the  left,  and  the  proper  abbreviations 
within  the  parentheses  : 

100  square  millimeters  =  1  square  centimeter     (sq.  cm) 
square  centimeters  =  1  square  decimeter      (  ) 

square  decimeters    =  1  square  meter  (  ) 

or  centar  (  ) 

square  meters  or     =1  square  dekameter     (  ) 

centars  or  ar  (  ) 

square  dekameters  =  1  square  hectometer  (  ) 

or  ars  or  hectar  (  ) 

square  hectometers  =  1  square  kilometer      (  ) 

or  hectars 

*  Pronounced  like  the  word  are.  This  and  our  word  area  are 
derived  from  the  Latin  area,  a  broad,  level  piece  of  ground. 


B.]  The  Metric  System.  269 

10.  How  many  square  meters  or  centars  are  there  in. 
3  ha  4  a  2  ca  ? 

11.  Change  684.2  sq.  m  to  ars  ;  to  square  decimeters. 

12.  Since,  in  square  measure,  each  measure  or  unit  is 
100  times  the  next  smaller,  how  many  places  must  we 
move  the  decimal  point  when  we  change  from  one  meas- 
ure to  the  next  ? 

13.  Reduce  398420  ars  to  hectars  ;  to  square  kilome- 
ters ;  to  centars ;  to  square  centimeters. 

14.  Reduce  2.246  square  kilometers  to  centars. 

15.  How  many  ars  are  there  in  a  barn-yard  89  dm 
long  and  7  m  wide  ? 

16.  How  many  square  kilometers  are  there  in  a  field 
3100  m  wide  and  100  Hm  long  ? 

17.  How  many  square  meters  of  carpet  would  he  re- 
quired to  carpet  the  room  that  you  are  now  in  ? 

18.  How  many  square  centimeters  are  there  in  the 
cover  of  this  book  ? 

19.  How  many  square  meters  of  paper  are  needed 
for  300  pages  of  this  book  ? 

20.  How  many  meters  of  carpet  .6  m  wide  are  needed 
to  carpet  a  room  10.8  m  long  and  4.6  m  wide  ? 

21.  Complete  the   following   table   by  filling  in  the 
proper  numbers  on  the  right,  and  the  proper  abbrevia- 
tions within  the  parentheses  :  — 

1  sq.  millimeter  (sq.  mm)  =  .000001  sq.  meter. 

1  sq.  centimeter  (             )  sq.  meter. 

1  sq.  decimeter  (             )  sq.  meter. 

1  centar  (       )  or  sq.  meter  (  )  =                 sq.  meter. 

1  ar  (      )  or  sq.  dekameter  (  )  =                 sq.  meters. 

1  hectar  (    )  or  sq.  hectometer  sq.  meters. 

1  sq.  kilometer  (       )  sq.  meters. 


270 


Appendix. 


[Ch.  2. 


C.  Solid  Measure. 

A  cubic  centimeter  (cu.  cm)  is  a  solid,  each  of  whose 
edges  is  a  centimeter. 


1.  How  many  cubic  centimeters  are  there  in  a  solid 
2  cm  long,  1  cm  wide,  and  1  cm  high  ?  how  many  in  a 
solid  4  cm  long,  1  cm  wide,  and  1  cm  high  ? 

2.  How  many  cubic    centimeters    are    there   in  the 
solids  whose  dimensions    (length,  breadth,  and   thick- 
ness) are  as  follows  :  — 

a.  10  cm,  2  cm,  and  1  cm 

b.  10  cm,  4  cm,  and  1  cm 

c.  10  cm,  10  cm,  and  1  cm 

d.  10  cm,  10  cm,  and  2  cm 

e.  10  cm,  10  cm,  and  10  cm. 

3.  How  many  cubic  centimeters  are  there  in  a  cubic 
decimeter  ? 

4.  How  many  cubic  decimeters   (cu.  dm)  are  there 
in  a  cubic  meter  ? 

[The  cubic  meter  (cu.  m)  is  also  called  a  ster  *  («).] 

5.  How  many  cubic  millimeters  (cu.  mm)  are  there 
in  a  cubic  centimeter  ? 

6.  Complete   the   following   table   by  filling   in   the 
proper  numbers  on  the  left : 

cubic  millimeters  =  1  cubic  centimeter 
cubic  centimeters  =  1  cubic  decimeter, 
cubic  decimeters  =  1  cubic  meter  or  ster. 
*  Pronounced  like  the  word  stair. 


D.]  The  Metric  System.  271 

7.  Reduce  16900  cubic  decimeters  to  sters;  to  cubic 
centimeters. 

8.  Reduce  3  5,  4  cu.  dm,  3  cu.  cm  to  cubic  centime- 
ters. 

9.  In  cubic  measure  each  measure  or  unit  is  1000 
times  the  next  smaller  ;  how  many  places,  then,  must 
we  move  the  decimal  point  when  we  change  from  one 
measure  to  the  next  smaller  or  larger  ? 

10.  Change  164300000  cu.  mm  to  cubic  centimeters ; 
change  the  result  to  cubic  decimeters  ;  change  the  last 
result  to  cubic  meters. 

11.  In  3.1647  sters  how  many  cubic  decimeters  are 
there  ?  how  many  cubic  centimeters  ? 

12.  How  many  cubic  meters  are  there  in  a  box  whose 
dimensions  are  4  m,  5  dm,  and  5  dm  ? 

13.  How  many  sters  are  there  in  a  pile  of  wood  8  m 
long,  1  in  wide,  and  2  m  high  ? 

14.  How  many  cubic  meters  are  there  in  the  room 
that  you  are  now  in  ? 

15.  How  many  cubic  decimeters  of  water  will  a  tank 
hold  that  is  2  m  long,  .5  m  wide,  and  .7  m  deep  ? 

1U/W 
D.  Capacity  Measure. 

1.  Draw  on  a  piece  of  card- 
board a  figure  like  that  given 
here,  making  each  of  the  lines 
1  decimeter  long  ;  cut  the  figure 
out  with  a  sharp  knife  ;  cut  the 
card-board  half  through  under- 
neath the  dotted  lines  ;  fold  up 
the  outer  portions  so  as  to  form 
a  box.  This  box  will  hold  a  cubic  decimeter  or  liter. 


272  Appendix.  [Ch.  2. 

NOTE.  The  liter  is  the  principal  measure  of  capacity  ; 
the  smaller  measures  of  capacity  are  one 
tenth,  one  hundredth,  and  one  thousandth 
of  a  liter ;  and  the  larger  measures  of  capa- 
city are  ten,  one  hundred,  and  one  thousand 
liters.  The  names  of  the  different  measures 
of  capacity  are  formed  by  prefixing  to  the  word  liter 
the  same  names  that  are  used  as  prefixes  in  linear  meas- 
ure. 

2.  Form  a  table  of  the  measures  of  capacity  like  that 
of  the  measures  of  length. 

3.  How  many  liters  will  a  tank  hold  that  is  2  dm 
wide,  5  dm  long,  and  6  cm  high  ? 

E.   Weight  Measure. 

The  weight  of  the  water*  that  can  be  put  into  a  tight 
box  of  the  size  of  the  one  just  mentioned,  that  is,  the 
weight  of  a  liter  or  cubic  decimeter  of  water  is  called 
a  kilogram  (kg)  or,  more  briefly,  a  kilo. 

1.  How  many  kilograms  will  5  liters  of  water  weigh  ? 
how  many  kilos  will  a  hectoliter  of  water  weigh  ? 

2.  How  many  grams  (g)  will  a  cubic  centimeter  of 
water  weigh  ?  how  many  kilograms  will  a  cubic  meter 
of  water  weigh  ? 

3.  Form  a  table  of  the  measures  of  weight  like  that 
of  the  measures  of  length. 

4.  The  weight  of  a  cubic  meter  of  water  (1000  kg) 
is  called  a  metric  ton.     How  many  metric  tons  of  water 
will  a  tank  hold  that  is  3  m  long,  1.5  m  wide,  and  .5  m 
deep  ? 

*  Pure  water  taken  at  its  greatest  density,  that  is,  a  little  above 
the  freezing  point. 


F.]  The  Metric  System.  273 

F.  Miscellaneous. 

Examples. 

1.  Reduce  2.15  km  to  centimeters. 

2.  Reduce  2.15  miles  to  inches. 

3.  How  many  hectars  are  there  in  a  rectangular  park 
1  km  26  m  long,  and  675  m  wide  ? 

4.  How  many  acres  are  there  in  a  rectangular  park 
1  mile  26  yds.  long  and  675  yds.  wide  ? 

5.  Reduce  6  Dm  to  decimeters. 

6.  Reduce  6  gallons  to  gills. 

7.  Reduce  6  metric  tons  to  grams. 

8.  Reduce  6  tons  to  ounces. 

9.  Examples  1,  3,  5,  and  7  are  like  Examples  2,  4, 
6,  and  8,  except  that  in  the  former  the  French  or  metric 
system  of  measures  is  used,  and  in  the  latter  the  Eng- 
lish system  of  measures  is  used.     Which  set  of  exam- 
ples can  be  solved  in  the  least  time  and  with  the  least 
amount  of  figuring  ? 

As  we  have  seen,  there  are  the  following  simple  rela- 
tions between  the  metric  measures  of  length,  weight, 
and  capacity :  A  CUBIC  DECIMETER  is  A  LITER,  AND  A 

LITER    OF   WATER    WEIGHS    A    KILOGRAM  ;     whereas    in 

the  common  system  of  measures  there  are  no  simple  re- 
lations between  the  corresponding  measures,  the  foot  (or 
inch),  the  pound,  and  the  quart. 

In  the  entire  metric  system  the  only  words  required 
to  designate  the  different  measures  are  meter,  liter, 
gram,  metric  ton,  ar,  ster,  and  the  prefixes  milli,  centi, 
deci,  deka,  Tiecto,  kilo,  and  myria  ;  whereas  in  the  com- 
mon system,  in  addition  to  numerous  words  occasionally 
required,  the  following  21  are  in  common  use  :  inch, 


274  Appendix.  [Ch.  2. 

foot,  yard,  rod,  mile,  acre,  cord  foot,  cord,  gill,  pint, 
quart,  gallon,  peck,  bushel,  grain,  scruple,  dram,  ounce, 
pennyweight,  pound,  ton. 

The  name  of  a  metric  measure  tells  at  once  how 
many  principal  units  it  contains  ;  the  name  kilometer, 
for  instance,  means  1000  meters  ;  whereas  the  name 
mile  is  no  indication  of  the  number  of  feet. 

From  what  has  been  said,  we  see  that  if  the  metric 
system  of  measures  were  everywhere  substituted  for  the 
common  system,  a  great  saving  of  time  and  labor  ivould 
be  made  both  in  the  education  of  children  and  in  the 
necessary  computations  of  every-day  life  : 

1.  Because,  as  it  is  a  decimal  system,  a  change  from 
one  denomination  to  another  can  be  made  by  a  mere 
change  in  the  position  of  the  decimal  point. 

2.  Because  of  the  simple  and  easily  remembered  re- 
lations which   exist   between   the  measures  of  length, 
capacity,  and  weight. 

3.  Because  of  the  small  number  of  words  required  to 
designate  all  the  different  measures. 

4.  Because  almost  every  word  used  has  a  meaning 
that  tells  the  size  of  the  measure  to  which  it  belongs. 

Professor  LEONE  LEVI,  in  his  Metric  System,  says :  — 

' '  Here  is  a  tool  which  offers  facilities  for  saving  one  half  the 
time  in  arithmetical  education,  and  one  fourth,  or  one  third  of  the 
time  spent  in  all  the  transactions  which  include  calculations  of 
weights  and  measures." 

The  following  are  extracts  from  a  paper  by  Edward  Wiggles- 
worth,  M.  D.  :  — 

"John  Quincy  Adams,  even  in  his  day,  spoke  of  the  metric 
system  as  '  the  greatest  invention  of  human  ingenuity  since  that  of 
printing.'  It  has  been  calculated  by  large  committees  of  our 
ablest  teachers  that  the  complete  introduction  of  the  metric  system 
will  save  a  full  year  of  the  school-life  of  every  child  .  .  .  simpler 
than  others  as  our  money  is  simpler  than  pounds,  shillings,  and 


Arithmetical  Tables.  275 

pence,  .  .  .  competent  authorities  computed  that  the  London  and 
Northwestern  Railway  alone  would  annually  save  £10,000  sterling 
by  the  use  in  all  its  computations  of  the  metric  instead  of  the  old 
system.  How  vast,  then,  would  be  the  saving  in  the  entire  busi- 
ness of  the  country  !  In  1860  the  foreign  business  of  the  United 
States  equalled  $762,000,000.  Of  this  $700,000,000  was  with 
nations  using  the  metric  system,  and  that,  too,  before  Germany 
had  adopted  it." 


APPENDIX.     CHAPTER  III. 
ARITHMETICAL   TABLES. 

[Measures  less  frequently  used  are  printed  in  smaller 

type.] 

UNITED  STATES  MONEY. 

10  mills  =  1  cent. 
10  cents  =  1  dime. 
10  dimes  =  1  dollar  ($). 
$10  =  1  eagle. 

ENGLISH  MONEY. 
4  farthings  (f)  =  1  penny  (d.). 
12  pence  =1  shilling  (s.). 

20  shillings         =  1  pound  sterling  (£). 

A  florin  =  2  s.  A  sovereign  =  20  s. 

A  crown  =  5  s.  A  guinea      =  21  s. 

1£  =  $4.866  J. 

FRENCH  MONEY. 
100  centimes  =  1  franc  ($0.193). 

GERMAN  MONEY. 
100  pfennigs  =  1  mark  ($0.238). 


276  Appendix.  [Ch.  3. 

LENGTH. 
12  inches  (in.)^l  foot  (ft.). 

3  feet  =  1  yard  (yd.). 

16£  feet  or) 

5*  yards   I       = 
320  rods  or) 
5280  feet       j 

12  lines    =  1  inch.  18  inches     =  1  cubit. 

4  inches  =  1  hand.  40  rods         =  1  furlong. 

9  inches  =  1  span.  S  furlongs  —  1  mile. 

6  feet  =  1  fathom. 

SURVEYORS'  MEASURE  OF  LENGTH. 
7.92  inches  =1  link  (Ink.). 
100  links    =1  chain  (ch.). 
80  chains  =  1  mile. 
25  Inks  =  1  rd. 

The  Surveyors'  Measure  was  obtained  by  first  calling 
one  tenth  of  an  acre  *  a  square  chain  and  then  divid- 
ing a  linear  chain  (4  rds  or  792  in.)  into  100  equal 
parts  which  were  called  links. 

SURFACE. 

144  square  inches  =:1  square  foot  (sq.  ft.). 
9  square  feet          =  1  square  yard  (sq.  yd.). 

272J  square  feet     ) 

0   *  ,    \    —\  square  rod  (sq.  rd.). 

or  30J  square  yards  ) 

160  square  rods         =1  acre  (a.). 

640  acres  =1  square  mile  (sq.  m.). 

40  sq.  rds.  =  1  rood.  1  sq.  m.    =1  section. 

4  roods     =  1  acre.  36  sections  =  1  township. 


*  One  tenth  of  an  acre,  or  16  sq.  rds.,  is  equivalent  to  a  square 
each  side  of  which  is  4  rds.  If,  then,  16  sq.  rds.  are  called  a 
square  chain,  a  linear  chain  would  be  4  rds.  or  792  in. 


Arithmetical  Tables.  277 

SOLIDITY. 

1728  cubic  inches  (cu.  in.)  =  l  cubic  foot  (cu.  ft.). 
27  cubic  feet  =1  cubic  yard  (cu.  yd.). 

16  cubic  feet  =  1  cord  foot. 

8  cord  feet  or  ) 

HOO      i  •    £  =1  cord  (cd.). 

128  cubic  feet     ) 

MEASURES  OF  CAPACITY. 
LIQUID   MEASURE. 

4  gills     =1  pint  (pt.). 
2  pints    =1  quart  (qt.). 
4  quarts  =  1  gallon  (gal.). 

DRY   MEASURE. 

2  pints  =1  quart  (qt.). 
8  quarts  =  1  peck  (pk.). 
4  pecks  =1  bushel  (bu.). 

4  quarts  liquid  measure  =  231  cubic  inches. 
4  quarts  dry  measure       =  26&|  cubic  inches. 

AVOIRDUPOIS  WEIGHT. 

FOR    ALL   GOODS    EXCEPTING    GOLD,  SILVER,    AND 
PRECIOUS    STONES. 

16  drams  (dr.)  =  l  ounce  (oz.). 
16  ounces  =  1  pound  (lb.)0 

2000  pounds          =  1  ton  (t). 

14  pounds  =  1  stone. 
100  pounds  =  1  hundred-weight  (cwt.). 
2240  pounds  ==  1  long  ton.. 

The  long  ton  is  used  in  the  United  States  custom- 
house and  in  wholesale  transactions  in  coal  and  iron. 


278  Aj,jx'nflix.  [Ch.  3. 

TROY  WEIGHT.    . 
FOB   GOLD,    SILVER,    AND    PRECIOUS   STONES. 

24  grains  (grs.)  =1  pennyweight  (dwt.). 
20  penny  weights  =  1  ounce  (oz.). 
12  ounces  =1  pound  (lb.). 

1  lb.  Troy  =  5700  grs. 

1  lb.  Avoirdupois  =  7000  grs. 

1  carat  of  diamond          =      4  grs. 
1  carat  of  gold  or  silver  =  240  grs. 
24  carats  of  gold  or  silver  =  1  lb. 

Pure  gold  is  too  soft  for  practical  use  ;  when  coins,  jewels,  etc., 
are  said  to  be  made  of  gold  they  really  consist  of  a  mixture  of 
pure  gold  and  some  other  metal.  "Jewelers'  gold,"  a  pound 
('J4  carats)  of  which  contains  only  18  carats  of  pure  gold,  is  said 
to  be  18  carats  fine.  "  Standard  gold  "  is  22  carats  fine  ;  that  is, 
in  every  24  parts  there  are  22  parts  of  pure  gold  and  2  parts  of 
some  other  metal.  On  the  inside  of  a  gold  watch  case  or  of  a 
gold  ring  we  usually  find  a  stamp  to  indicate  the  fineness  of  the 
gold,  such  as  18  k  or  merely  18. 

APOTHECARIES'   WEIGHT. 

20  grains  (grs.)  =  l  scruple  (3). 
3  scruples          =1  dram  (3). 
8  drams  =  1  ounce  (oz.  or  |). 

12  ounces  =  1  pound  (lb.). 

The  grain,  ounce,  and  pound  are  the  same  as  in  Troy  weight. 

APOTHECARIES'   MEASURE. 


60  minims  (rn.)  =1  dram 

8  drams  =1  ounce  (fl.  drm.  viij.). 

16  ounces  =1  pint  (fl.  oz.  xvj.). 

HI  lx.  means  Ix.  (sixty)  minims  ;  fl.  drm.  viij.  means  viii. 
fluid  drams  ;  and  fl.  oz.  xvj.  means  xvi.  fluid  ounces.  The  j  at 
the  end  is  used  instead  of  an  i. 


Arithmetical  Tables.  279 


TIME. 

60  seconds  (sec.)  —  1  minute  (min.) 
60  minutes  =1  hour  (hr.J. 

24  hours  =1  day  (dy.). 

7  days  =1  week  (wk.). 

365  days  or       ) 

52  wk8.  i  dy.  }     common 

366  days  =  1  leap  year. 
100  years                =  1  century. 

Thirty  days  hath  September, 
April,  June,  and  November. 
All  the  rest  have  thirty-one, 
Excepting  February  alone, 
To  which  we  twenty-eight  assign, 
Till  leap  year  gives  it  twenty-nine. 

A  solar  year  is  365  dys.  5  hrs.  48  min.  50  sec.,  that  is,  nearly 
365£  days.  A  common  year  of  365  days  is,  therefore,  nearly  one 
fourth  of  a  day  shorter  than  a  solar  year  ;  to  make  up  for  this 
defect  some  years  (leap  years)  are  reckoned  as  366  days.  When- 
ever the  number  representing  the  year  is  divisible  by  4  and  not 
by  100,  or  whenever  it  is  divisible  by  400,  then  the  year  is  a  leap 
year.  Thus  1888  is  a  leap  year  ;  1889  is  not  a  leap  year  ;  1900 
is  not  a  leap  year  ;  2000  is  a  leap  year. 

MISCELLANEOUS. 

NUMBERS.  PAPER. 

12  units  =1  dozen.  24  sheets    =1  quire. 

12  dozen  =  1  gross.  20  quires   =  1  ream. 
12  gross  =  1  great  gross.  2  reams    =  1  bundle. 

20  units  =  1  score.  5  bundles  =  1  bale. 

A  barrel  of  flour  =  196  Ibs.          A  cask  of  lime     =  240  Ibs. 
Hr  A  barrel  of  pork  =  200  Ibs.          A  quintal  of  fish  =  100  Ibs. 
or  beef 


280  Appendix. 

THE  METRIC  SYSTEM  OF  MEASURES. 
LINEAR   MEASURE. 

10  millimeters    (mm)  =  1  centimeter  (cm). 

10  centimeters  =  1  decimeter  (dm). 

10  decimeters  =  1  meter  (m). 

10  meters  =  1  dekameter  (Dm). 

10  dekameters  =  1  hectometer  (Hm). 

10  hectometers  =1  kilometer  (Km). 

10  kilometers  =1  myriameter  (Mm). 

1  meter  =  39.3707904  in.  =  3.28090  ft.  =  1.09363  yds. 
1  kilometer  =  0.62138  miles. 

SQUARE   MEASURE. 

100  square  millimeters  (sq.  mm) 

=  1  square  centimeter  (sq.  cm). 

100  square  centimeters  =  1  square  decimeter  (sq.  dm). 
100  square  decimeters     =  1  square  meter  (sq.  m) 

or  centar  (ca). 
100  square  meters  =  1  square  dekameter  (sq.  Dm) 

or  ar  (a). 
100  square  dekameters    =  1  square  hectometer  (sq.  Hm) 

or  hectar  (Ha). 
100  square  hectometers  =  1  square  kilometer  (sq.  Km). 

1  sq.  m  =  1.19603  sq.  yds.  =  10.76430  sq.  ft. 
1  ar  =  3.95383  sq.  rds. 

CUBIC   MEASURE. 

1000  cubic  millimeters  (cu.  mm) 

=  1  cubic  centimeter  (cu.  cm), 

1000  cubic  centimeters     =1  cubic  decimeter  (cu.  dm). 
1000  cubic  decimeters        =  1  cubic  meter  (cu.  m) 

or  ster  (s.). 
1  cu.  m.  =  35.31658  cu.  ft.  =  1.30802  cu.  yds. 


Arithmetical  Tables.  281 

CAPACITY  MEASURE. 

10  milliliters    (ml)  =  1  centiliter  (cl). 
10  centiliters  =  1  deciliter  (dl). 

10  deciliters  =1  liter  (I). 

10  liters  =1  dekaliter  (Dl). 

10  dekaliters  =1  hectoliter  (HI). 

10  hectoliters          =1  kiloliter  (Kl). 
1  /  =  1  cu.  dm  = 

=  1.0567  quarts  liquid  measure. 

=  .90792  quarts  dry  measure. 

WEIGHTS. 

10  milligrams  (ing)  =  1  centigram  (eg). 
10  centigrams  =1  decigram  (dg). 

10  decigrams  =1  gram  (g). 

10  grams  =1  dekagram  (Dg). 

10  dekagrams  =  1  hectogram  (ffg)» 

10  hectograms  =1  kilogram  (K). 

1000  kilograms  =1  metric  ton  (T).   

1  I  or  1  CM.  dm  of  water  weighs  1  K. 

1  K.  =  2.20462  Ibs.  Avoirdupois. 
=  2.67923  Ibs.  Troy. 

Of  the  coins  of  the  United  States 

The  silver  half-dollar  weighs  12J  grams. 

"         "      quarter-dollar         weighs    6£  grams. 
"         "      twenty-cent  piece  weighs    5    grams, 
ten-cent  piece         weighs    2i  grams. 
"      nickel  five-cent  piece       weighs    5    grams. 


282  Compound  Interest  Table. 

COMPOUND  INTEREST  TABLE. 

SHOWING  THE  AMOUNT  OF  $1.00,    AT   COMPOUND   INTEREST, 
FROM  1  YEAR  TO  50. 


Year. 

3  p.  cent. 

3£  p.cent. 

4  p.  cent. 

4Jp.cent. 

5  p.  cent. 

6  p.  cent. 

7  p.  cent. 

1 
2 
3 
4 
5 

1.030000 
l.OGOOOO 
1.092727 
1.125509 
1.159274 

1.035000 
1.071225 
1.108718 
1.147523 
1.187686 

1.040000 
1.081600 
1.124864 
1.169859 
1.216653 

1.045000 
1.092025 
1.141166 
1.192519 
1.246182 

1.050000 
1.102500 
1.157625 
1.215506 
1.276282 

1.060000 
1.123600 
1.191016 
1.262477 
1.338226 

1.070000 
1.144900 
1.225043 
1.3107% 
1.402552 

6 

7 
8 
9 
10 

1.194052 
1.229874 
1.266770 
1.304773 
1.343916 

1.229255 
1.272279 
1.316809 
1.362897 
1.410599 

1.265319 
1.315932 
1.368569 
1.423312 
1.480244 

1.302260 

1.360862 
1.422101 
1.486095 
1.  5521)60 

1.3400% 
1.407100 
1.477455 
.551328 

.628895 

1.418519 
1.503630 
1.593848 
1.689479 
1.790848 

1.500730 
1.605781 
1.718186 
1.838459 
1.967151 

11 
12 
13 
14 
15 

1.384234 
1.425761 
1.468534 
1.512590 
1.557967 

1.459970 
1.511069 
1.5631)56 
1.618694 
1.675349 

1.539454 
1.601032 
1.665073 
1.731676 

1.800043 

1.622853 
1.695881 
1.7721% 
1.851945 
1.935282 

.710339 
.795856 

.885649 
1.979932 

2.078928 

1.898299 
2.0121% 
2.132928 
2.260904 
2.39655S 

2.104862 
2.252192 
2.409845 
2.578534 
2.759031 

16 
17 
18 
19 
20 

1.604706 
1.652848 
1.702433 
1.753506 
1.806111 

1.733986 
1.794675 

1.857480 
1.1)22501 
1.989789 

1.872981 
1.947900 

2.025816 
2.106849 
2.191123 

2.022370 
2.113377 

2.208479 
2.307860 
2.411714 

2.182875 
2.292018 
2.406619 

2.526050 
2.653298 

2.540352 
2.692773 

2.854:53') 
3.02551*0 
3.207135 

2.952164 
3.158815 
3.379931 
3.616526 

3.800683 

21 
22 
23 
24 
25 

1.860295 
1.916103 
1.973586 
2.032794 
2.093778 

2.059431 
2.131512 
2.206114 
2.283328 
2.363245 

2.278768 
2.369919 
2.464715 
2.563304 
2.665836 

2.520241 
2.  (33652 
2.752166 
2.876014 
3.005434 

2.785963 
2.025261 
3.071524 
3.225100 
3.386355 

3.399564 
3.603537 
3.819750 
4.048935 
4.291871 

4.140561 
4.430400 
4.740528 
5.072365 
5.427431 

26 
27 
28 
29 
30 

2.156591 
2.221289 
2.287928 
2.356565 
2.427262 

2.445959 
2.531567 
2.620177 
2.711878 
2.806794 

2.772470 
2.883300 
2.998703 
3.118651 
3.243397 

3.140679 
3.282009 
3.429700 
3.584030 
3.745318 

3.555673 
3.733456 
3.920129 
4.116136 
4.321942 

4.549383 
4.822346 
5.111687 
5.418388 
5.743491 

5.807351 
6.213866 
6.648836 
7.114255 
7.612253 

31 
32 
33 
34 
35 

2.500080 
2.575083 
2.652335 
2.731905 
2.813862 

2.905031 
3.006708 
3.111942 
3.220860 
3.333590 

3.373133 
3.508059 
3.648381 
3.794316 
3.946089 

3.913857 

4.089981 
4.274030 
4.466361 
4.667348 

4.538039 
4.764941 
5.003188 
5.253348 
5.516015 

6.088101 
6.453387 
6.840590 
7.251025 
7.686087 

8.145110 
8.715268 
9.325337 
9.978110 
10.676578 

36 
37 

38 
39 
40 

2.898278 
2.985227 
3.074783 
3.167027 
3.262038 

3.450266 
3.571025 
3.696011 
3.825372 
3.959260 

4.103932 
4.268090 
4.438813 
4.616366 
4.801021 

4.877378 
5.096860 
5.326219 
5.565899 
5.816364 

5.791816 
6.081407 
6.385477 
6.704751 
7.039989 

8.147252 
8.636087 
9.154252 
9.703507 
10.285718 

11.423939 
12.223614 
13.079277 
13.994827 
14.974465 

41 
42 
43 
44 
45 

3.359899 
3.4606% 
3.564517 
3.671452 
3.781596 

4.097834 
4.241258 
4.389702 
4.543342 
4.702358 

4.993061 
5.192784 
5.400495 
5.616515 
5.841176 

6.078101 
6.351615 
6.637438 
6.936123 

7.248248 

7.391988 
7.761587 
8.149667 
8.557150 
8.985008 

10.902861 
11.557033 
12.250455 
12.985482 
13.764611 

16.022677 
17.144265 
18.344363 
19.628469 
21.002461 

46 
47 
48 
49 
50 

3.895044 
4.011895 
4.132252 
4.256219 
4.383906 

4.866941 
5.037284 
5.213589 
5.396065 
5.584927 

6.074823 
6.317816 
6.570528 
6.833349 
7.106683 

7.574420 
7.915268 
8.271455 
8.643671 
9.0326361 

9.434258 
9.905971 
10.401270 
10.921333 
11.467400 

14.590487 
15.465917 
16.393872 
17.377504 
18.420154 

22.472634 
24.045718 
25.728918 
27.529943 
29.457039 

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rather  than  with  the  author. 

The  numbers  already  issued  have  been  extensively  used  for  the  study 
of  Language,  for  the  study  of  Literature,  for  Supplementary  Reading, 
and  as  substitutes  for  the  graded  Readers.  In  whatever  way  they  may 
be  used,  the  principal  benefit  to  be  derived  from  them  will  be  the  forma- 
tion of  a  taste  in  the  reader  for  the  best  and  most  enduring  literature ; 
this  taste  the  pupil  will  carry  with  him  when  he  leave:  school,  and  it  will 
remain  through  life  a  powerful  means  of  self-education.  An  inspection 
of  the  titles  of  the  different  numbers  of  the  series  will  show  at  it  con- 
tains a  pleasing  variety  of  reading  matter  in  Biography,  History,  Poetry, 
and  Mythology. 

While  each  number  has  been  issued  in  paper  covers,  several  combinar 
tions  of  two  and  three  numbers  in  board  covers  have  also  been  made,  in 
response  to  a  demand  for  a  larger  amount  of  material  in  a  single  volume 
with  a  more  permanent  binding. 

The  Publishers  take  pleasure  in  announcing  that  several  new  num- 
bers—containing some  of  the  best  and  purest  literature  —  will  be 
added  to  the  Riverside  Literature  Series  during  each  school  year ;  the 
wide-spread  popularity  among  teachers  and  pupils  of  the  numbers 
a'.oady  published  is  a  sufficient  guarantee  that  future  numbers  wil 
meet  with  favor. 

GRADING. 

Numbers  47,  48,  49,  and  50  are  suitable  for  pupils  of  the  Second 
and  Third  Reader  grades.  The  following:  numbers,  given  in  the 
order  of  their  simplicity,  have  been  found  well  adapted  to  tKe 
tastes  and  capabilities  of  pupils  of  the  Fourth  Reader  grades 
29,  10,  7,  8,  9,  17,  18,  22,  23,  46,  11,  21,  44,  28,  36,  24, 19,  20f 
32, 37, 31,  F,  G,  and  H.  The  other  numbers  of  the  series  are  suita- 
ble for  pupils  of  the  Fifth  and  Sixth  Reader  grades  and  for  the 
study  of  literature. 

1  There  are  in  the  entire  series  perhaps  half  a  dozen  cases  where  a  sentence  has 
been  very  flight]/  changed  in  order  to  adapt  it  for  use  in  the  schoolroom ;  and  in 
•to  tvN,  lor  similar  reasons-  three  page0«f  the  original  have  been  omitted. 


*  C^e  IBtoersilie  literature  Aeries. 

With  Introductions,  Notes,  Historical  Sketches,  and  Biographical  Sketches. 
Each  single  number  in  paper  covers,  15  cents. 

1.  Longfellow's  Evangeline. 

2.  Longfellow's  Courtship  of  Miles  Standish. 

3.  Longfellow's  Courtship   of  Miles   Standish.     DRAMATIZED 

for  private  theatricals  in  schools  and  families. 

4.  Whittier's  Snow-Bound,  Among  the  Hills,  and  Songs  of 

Labor. 

5.  Whittier's  Mabel  Martin,  Cobbler  Keezar,  Maud  Mullerr 

and  Other  Poems. 
.  Holmes's  Grandmother's  Story  of  Bunker  Hill  Battle,  and 

Other  Poems. 

7,  8,  9.  Hawthorne's  True   Stories  from  New  England  His- 
tory.    1620-1803.     In  three  parts.f 

10.  Hawthorne's  Biographical  Stories. 

Sir  Isaac  Newton,  Samuel  Johnson,  Oliver  Cromwell,  Benjamin 
Franklin,  Queen  Christina.     With  Questions. 

[29  and  10  also  in  one  volume,  board  covers,  40  cents.] 

11.  Longfellow's  Children's  Hour,  and  other  Selections. 

12.  Studies   in  Longfellow.     Containing   Thirty-Two   Topics    for 

Study,  with  Questions  and  References  relating  to  each  Topic. 
13, 14.  Longfellow's  Song  of  Hiawatha.     In  two  parts.f 

15.  Lowell's  Under  the  Old  Elm,  and  Other  Poems. 

16.  Bayard  Taylor's  Lars;    a  Pastoral  of  Norway. 
17, 18.  Hawthorne's  Wonder-Book.    In  two  parts.J 

19, 20.    Benjamin    Franklin's    Autobiography.     With  a  chaptei 
completing  the  Life.     In  two  parts.! 

21.  Benjamin  Franklin's  Poor  Richard's  Almanac,  and  othec 

Papers. 

22,  23.  Hawthorne's    Tanglewood    Tales.    In  two  parts.f 

24.  Washington's  Rules  of  Conduct,  Letters  and  Addresses. 

25,  26.  Longfellow's  Golden  Legend.    In  two  parts.:}: 

27.  Thoreau's  Succession  of  Forest  Trees,  and  Wild  Apples 

With  a  Biographical  Sketch  by  R.  W.  EMERSON. 

28.  John  Burroughs^  Birds  and  Bees. 

[28  and  36  also  in  one  volume,  board  covers,  40  cents.] 

29.  Hawthorne's  Little  Daffydowndilly,  and  other  Stories. 

[29  and  10  also  in  one  volume,  board  covers,  40  cents.] 

30.  Loweirs  Vision  of  Sir  Launf  al  and  Other  Pieces. 

31.  Holmes's  My  Hunt  after  the  Captain  and  Other  Papers. 

32.  Abraham  Lincoln's  Gettysburg  Speech,  and  Other  Papers, 

33.  34,  35.  Longfellow's  Tales  of  a  Wayside  Inn.    In  three  parts. 

[The  three  parts  also  in  one  volume,  board  covers,  50  cents.] 

36.  John  Burroughs's  Sharp  Eyes  and  otter  Papers. 

[28  and  36  also  in  one  volume,  board  covers,  40  cents.] 

37.  Charles  Dudley  Warner's  A-Hunting  of  the  Deer,  and 

Other  Papers. 

t  Also  in  one  volume,  board  covers,  45  cents, 
i  Also  in  one  volume,  board  covers,  40  cents. 

Continued  on  the  inside  of  this  cover. 

HOUGHTON,  MIFFLIN  AND  COMPANY, 

4  PARK  STREET,  BOSTON,  MASS. 


Literature 

[.4.  list  of  the  first  thirty-seven  numbers  is  given  on  the  back  cover."] 

38.  Longfellow's  Building  of  the  Ship,  Masque  of  Pandora, 

and  Other  Poems. 

39.  Lowell's  Books  and  Libraries,  and  Other  Papers. 

40.  Hawthorne's  Tales  of  the  White  Hills,  and  Sketches. 

41.  Whittier's  Tent  on  the  Beach. 

12.    Emerson's  Fortune  of  the  Republic,  and  Other  American 

Essays. 

43.  Ulysses  among  the  Phaeacians.    From  W.  C.  BRYANT'S  Trans- 

lation of  Homer's  Odyssey. 

44.  Maria  Edgeworth's  Waste  Not,  "Want  Not,  and  Barring 

Out. 

45.  Macaulay's  Lays  of  Ancient  Rome. 

46.  Old  Testament  Stories  in  Scripture  Language.    From  the 

Dispersion  at  Babel  to  the  Conquest  of  Canaan. 

47, 48.    Fables    and    Folk    Stories.      Riverside    Second    Reader. 

Phrased  by  HORACE  E.  SCUDDER.     In  two  parts.* 
49,  50.    Hans  Andersen's    Stories.    JSewly  Translated.    Riverside 

Second  Reader.     In  two  parts. $ 

51,  52.     "Washington  Irving :   Essays  from  the  Sketch  Book. 

[51.]  Rip  Van  Winkle  and  other  American  Essays.  [52.]  The  Voyage  and  other 
English  Essays.  In  two  parts.  J 

53.  Scott's  Lady  of  the  Lake.    Edited   by  W.  J.  ROLFE.    With 

copious  notes  and  numerous  illustrations.    (Double  number,  30  cents.) 

54.  Bryant's  Sella,  Thanatopsis,  and  Other  Foems. 

$  Also  in  one  volume,  board  covers,  40  cents. 

EXTRA  NUMBERS. 
A    American  Authors  and   their  Birthdays.    Programmes  ana 

Suggestions  for  the  Celebration  of  the  Birthdays  of  Authors.  With  a  Record  of 
Four  Years'  Work  in  the  Study  of  American  Authors.  By  ALFRED  S.  ROB, 
Principal  of  the  High  School,  Worcester,  Mass. 

S    Portraits  and  Biographical  Sketches  of  Twenty  American 

C    A  Longfellow  Night.     A  Short  Sketch  of  the  Poet's  Life,  witl? 

songs  and  recitations  from  his  works.    For  the  Use  of  Catholic  Schools  and 

Catholic  Literary  Societies.     By  KATHERINE  A.  O'KEEFFE. 
J>    Literature  in  School;  The  Place  of  Literature  in  Common  School 

Education ;  Nursery   Classics   in    School ;  American  Classics  in  School.     By 

HORACE  E.  SCUDDER. 
E    Harriet    Beecher    Stowe.     Dialogues  and    Scenes  from  Mrs. 

Stowe's  Writings.     Arranged  by  EMILY  WEAVER. 

F    Longfellow  Leaflets.  (Each,  a  Double  Number,  30  cents.) 

G    Whittier  Leaflets.  Poems  and  Prose   Passages  from 

H    Holmes  Leaflets.  the  Works  of  Longfellow,  Whittier, 

and  Holmes.   For  Reading  and  Recitation.  Compiled  by  JOSEPHINE  E.  HODGDON. 

Illustrated,  with  Introductions  and  Biographical  Sketches. 
t    The  Riverside   Manual  for  Teachers,  containing  Suggestions 

and  Illustrative  Lessons  leading  up  to  Primary  Reading.  By  I.  F.  HALL,  Super- 
intendent of  Schools  at  Arlington,  Mass. 


HOUGHTON,  MIFFLIN  AND  COMPANY, 
4  PARK  STREET,  BOSTON,  MASS. 


€4je  JBitocm&e  literature 


In  my  opinion  nothing  cultivates  the  reading  habit  like  putting  fresh 
and  interesting  books  into  the  hands  of  the  children.  I  think,  however, 
that  I  have  made  the  mistake  of  selecting  too  many  information  books. 
...  I  intend  to  try  the  experiment  of  introducing  more  of  the  classics 
of  our  noble  literature.  —  E.  A.  GASTMAN,  Supt.  of  Schools,  Decatur,  III. 

If  reading  is  carefully  taught,  we  do  not  need  the  Fifth  Reader  in  the 
ungraded  schools.  —  GEO.  R.  SHAWHAN,  Supt.  of  Schools,  Champaign 
Co.,  III. 

If  a  Fifth  Reader  is  dispensed  with,  aa  some  have  advised,  some, 
thing  as  good  or  better  mast  take  its  place,  such  as  supplementary 
reading  of  the  proper  grade,  consisting  of  good  selections  taken  from 
the  best  American  and  English  authors.  —  State  Course  of  Study  for  the 
Common  Schools  of  Illinois. 

The  Superintendent  of  Public  Instruction  for  the  State  of  Illinois  re- 
commends for  reading  in  the  Higher  Course  of  Study  for  the  Common 
Schools  of  Illinois  the  following  masterpieces  of  American  Authors, 
published  in  the  Riverside  Literature  Series :  — 

No.  2.   The  Courtship  of  Miles  Standish  .    .    .  Longfellow. 

No.  28.  Birds  and  Bees Burroughs. 

No.  4.   Snow-Bound Whktier. 

No.  31.  My  Hunt  after  the  Captain    ....  Holmes. 

No.  40.   The  Great  Stone  Face Hawthorne. 

No.  32.  Essay  on  Abraham  Lincoln    ....  LowelL 

The  following  numbers  of  the  Riverside  Literature  Series  have  been 
adopted  for  use  in  the  graded  and  ungraded  schools  of  Knox  County, 
Illinois,  as  a  part  of  the  course  in  Literature  for  1890-91 ;  Nos.  2, 
4,  14,  17,  and  81. 

Riverside  Literature  Series,  by  mail, paper  covers,  post-paid. 
Single  numbers,  each  .  •  15c.  10  or  more  at  one  time,  each  14c. 
100  or  more  books  selected  from  the  Series  at  one  time,  each  .  .  13c. 
Sample  sets  at  the  "  100  or  more  "  rate,  post-paid.  These  prices  are  net 

HOUGHTON,  MIFFLIN  AND  COMPANY, 

i  PARK  STREET,  BOSTON;  11  EAS$  I?TH  STREET,  NEW  YORK; 
28  LAKESIDE  BUILDING,  CHICAGO. 


YB 


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